Nonlinear Dynamics

References

40:335-336

Press; 1993

438-446

10

University Press; 1997

University Press; 1990

[1] Blümel GL, Reinhardt WP. Chaos in Atomic Physics. Cambridge: Cambridge

[2] Baker GL, Gollub JP. Chaotic Dynamics. Cambridge: Cambridge

Research Advances in Chaos Theory

[3] Berry MV. Quantum chaology, not quantum chaos. Physica Scripta. 1989;

[4] Devaney RL. A First Coarse in Chaotic Dynamical Systems. Reading,

[5] Ott E. Chaos in Dynamical Systems. Cambridge: Cambridge University

[6] Dirac PAM. The adiabatic invariance of the quantum integrals. Proceedings of the Royal Society. 1925;107:725-734

[7] Zhang W-M, Feng DH, Yuan J-M,

nonintegrability of quantum systems: Quantum integrability and dynamical symmetry. Physical Review A. 1989;40:

Wong S-J. Integrability and

MA: Addison-Wesley; 1992

Chapter 2

Abstract

question.

13

N-soliton solution, spectral data PACS: 00.30.Lk 02.30.Jr, 05.45.Yv

parameters is a relaxation process.

Loop-like Solitons

Dmitri B. Vengrovich

Vyacheslav O. Vakhnenko, E. John Parkes and

The physical phenomena that take place in nature generally have complicated

nonlinear features. A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. One remarkable feature of the VE is that it possesses loop-like soliton solutions. Loop-like solitons are a class of interesting wave phenomena, which have been involved in some nonlinear systems. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE). The VPE can be written in Hirota bilinear form. The Hirota method not only gives the N-soliton solution but enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering transform (IST) method. This method is the most appropriate way of tackling the initial value problem (Cauchy problem). The standard procedure for IST method is expanded for the case of multiple poles, specifically, for the double poles with a single pole. In recent papers some physical phenomena in optics and magnetism are satisfactorily described by means of the VE. The question of physical interpretation of multivalued (loop-like) solutions is still an open

Keywords: nonlinear evolution equations, solutions, Vakhnenko equation, Hirota method, Bäcklund transformation, inverse scattering problem,

From the nonequilibrium thermodynamic standpoint, models of a relaxing medium are more general than equilibrium models. To develop physical models for wave propagation through media with complicated inner kinetics, notions based on the relaxational nature of a phenomenon are regarded to be promising. Thermodynamic equilibrium is disturbed owing to the propagation of fast perturbations. There are processes of the interaction that tend to return the equilibrium. The parameters characterizing this interaction are referred to as the inner variables unlike the macroparameters such as the pressure p, mass velocity u and density ρ.

In essence, the change of macroparameters caused by the changes of inner

1. The high-frequency perturbations in a relaxing medium

#### Chapter 2

## Loop-like Solitons

Vyacheslav O. Vakhnenko, E. John Parkes and Dmitri B. Vengrovich

### Abstract

The physical phenomena that take place in nature generally have complicated nonlinear features. A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. One remarkable feature of the VE is that it possesses loop-like soliton solutions. Loop-like solitons are a class of interesting wave phenomena, which have been involved in some nonlinear systems. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE). The VPE can be written in Hirota bilinear form. The Hirota method not only gives the N-soliton solution but enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering transform (IST) method. This method is the most appropriate way of tackling the initial value problem (Cauchy problem). The standard procedure for IST method is expanded for the case of multiple poles, specifically, for the double poles with a single pole. In recent papers some physical phenomena in optics and magnetism are satisfactorily described by means of the VE. The question of physical interpretation of multivalued (loop-like) solutions is still an open question.

Keywords: nonlinear evolution equations, solutions, Vakhnenko equation, Hirota method, Bäcklund transformation, inverse scattering problem, N-soliton solution, spectral data PACS: 00.30.Lk 02.30.Jr, 05.45.Yv

#### 1. The high-frequency perturbations in a relaxing medium

From the nonequilibrium thermodynamic standpoint, models of a relaxing medium are more general than equilibrium models. To develop physical models for wave propagation through media with complicated inner kinetics, notions based on the relaxational nature of a phenomenon are regarded to be promising. Thermodynamic equilibrium is disturbed owing to the propagation of fast perturbations. There are processes of the interaction that tend to return the equilibrium. The parameters characterizing this interaction are referred to as the inner variables unlike the macroparameters such as the pressure p, mass velocity u and density ρ. In essence, the change of macroparameters caused by the changes of inner parameters is a relaxation process.

We restrict our attention to barotropic media. An equilibrium state equation of a barotropic medium is a one-parameter equation. As a result of relaxation, an additional variable ξ (the inner parameter) appears in the state equation

$$p = p(\rho, \xi) \tag{1}$$

∂2 p <sup>∂</sup>x<sup>2</sup> � <sup>c</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86583

Klein-Gordon equation.

<sup>x</sup> � cf <sup>t</sup> � �,~<sup>t</sup> <sup>¼</sup>

The constant <sup>α</sup> <sup>¼</sup> <sup>β</sup><sup>f</sup> <sup>=</sup> ffiffiffiffiffiffi

x~ ¼

ffiffiffi γf 2 q

Loop-like Solitons

follow this name.

Korteweg-de Vries equation:

<sup>β</sup><sup>f</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>

ffiffiffi γf 2 q

> ∂ ∂x

> > 2γf

tive term has the form of the nonlinear equation [14, 15]: ∂ ∂x

> ∂ ∂x

tion of the propagation of high-frequency perturbations.

2. Loop-like stationary solutions

independent variables η and τ defined by

considerably easier to solve.

15

∂u ∂t þ u ∂u <sup>∂</sup><sup>x</sup> � <sup>β</sup> <sup>∂</sup><sup>3</sup>

cf <sup>t</sup>, <sup>u</sup><sup>~</sup> <sup>¼</sup> <sup>α</sup>fc<sup>2</sup>

∂ ∂t þ u ∂ ∂x � �<sup>u</sup> <sup>þ</sup> <sup>α</sup> <sup>∂</sup><sup>u</sup>

> ∂ ∂t þ u ∂ ∂x

Historically, (9) has been called the Vakhnenko equation (VE), and we will

We note that (9) follows as a particular limit of the following generalized

� �

derived by Ostrovsky [16] to model small-amplitude long waves in a rotating fluid (γu is induced by the Coriolis force) of finite depth. Subsequently, (9) was known by different names in the literature, such as the Ostrovsky-Hunter equation, the short-wave equation, the reduced Ostrovsky equation, and the Ostrovsky-Vakhnenko equation depending on the physical context in which it is studied. The consideration here of (9) has interest from the viewpoint of the investiga-

The travelling wave solutions are solutions which are stationary with respect to

For the VE (9) it is convenient to introduce a new dependent variable z and new

1=2

, <sup>τ</sup> <sup>¼</sup> t vj j<sup>1</sup>=<sup>2</sup>

, (11)

a moving frame of reference. In this case, the evolution equation (a partial differential equation) becomes an ordinary differential equation (ODE) which is

z ¼ ð Þ u � v =∣v∣, η ¼ ð Þ x � vt =j j v

�2 f ∂2 p <sup>∂</sup>t<sup>2</sup> <sup>þ</sup> <sup>α</sup>fc

<sup>f</sup> � <sup>c</sup><sup>2</sup> e τpc<sup>2</sup> ecf

2 f ∂2 p2 <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>f</sup>

Equation (6) with (β<sup>e</sup> ¼ 0) is the well-known the Korteweg-de Vries (KdV) equation. The investigation of the KdV equation in conjunction with the nonlinear Schrodinger (NLS) and sine-Gordon equations gives rise to the theory of solitons [4–13]. We focus our main attention on (7). It has a dissipative term β<sup>f</sup> ∂p=∂x and a dispersive term γ<sup>f</sup> p. Without the nonlinear and dissipative terms, we have a linear

<sup>f</sup> � <sup>c</sup><sup>4</sup> e

2τ<sup>2</sup> pc4 e c2 f :

Let us write down (7) in dimensionless form. In the moving coordinate system with velocity cf , after factorization the equation has the form in the dimensionless variables

∂x

u ∂x<sup>3</sup>

p is always positive. Equation (8) without the dissipa-

� �<sup>u</sup> <sup>þ</sup> <sup>u</sup> <sup>¼</sup> <sup>0</sup>: (9)

, <sup>γ</sup><sup>f</sup> <sup>¼</sup> <sup>c</sup><sup>4</sup>

∂p ∂x

þ γ<sup>f</sup> p ¼ 0,

<sup>f</sup> p (tilde over variables x~,~t and u~ is omitted)

þ u ¼ 0: (8)

¼ γu (10)

(7)

and defines the completeness of the relaxation process. There are two limiting cases with corresponding sound velocities:

i. Lack of relaxation (inner interaction processes are frozen) for which ξ ¼ 1:

$$p = p(\rho, \mathbf{1}) \equiv p\_f(\rho), \quad c\_f^2 = d p\_f / d \rho; \tag{2}$$

ii. Relaxation which is complete (there is local thermodynamic equilibrium) for which ξ ¼ 0:

$$p = p(\rho, 0) \equiv p\_e(\rho), \quad c\_e^2 = d p\_e / d \rho. \tag{3}$$

Slow and fast processes are compared by means of the relaxation time τp.

To analyse the wave motion, we use the following hydrodynamic equations in Lagrangian coordinates:

$$\frac{\partial V}{\partial t} - \frac{1}{\rho\_0} \frac{\partial u}{\partial \mathbf{x}} = \mathbf{0}, \qquad \frac{\partial u}{\partial t} + \frac{1}{\rho\_0} \frac{\partial p}{\partial \mathbf{x}} = \mathbf{0}. \tag{4}$$

The following dynamic state equation is applied to account for the relaxation effects:

$$
\pi\_p \left( \frac{dp}{dt} - c\_f^2 \frac{d\rho}{dt} \right) + (p - p\_\epsilon) = 0. \tag{5}
$$

Here <sup>V</sup> � <sup>ρ</sup>�<sup>1</sup> is the specific volume and <sup>x</sup> is the Lagrangian space coordinate. Clearly, for the fast processes ωτ<sup>p</sup> ≫ 1 , we have relation (2), and for the slow ones ωτ<sup>p</sup> ≪ 1 , we have (3).

The closed system of equations consists of two motion equations (4) and dynamic state equation (5). The motion equations (4) are written in Lagrangian coordinates since the state equation (5) is related to the element of mass of the medium.

The substantiation of (5) within the framework of the thermodynamics of irreversible processes has been given in [1, 2]. We note that the mechanisms of the exchange processes are not defined concretely when deriving the dynamic state equation (5). In this equation the thermodynamic and kinetic parameters appear only as sound velocities ce and cf and relaxation time τp. These are very common characteristics and they can be found experimentally. Hence, it is not necessary to know the inner exchange mechanism in detail.

Combining the relationships (4) and (5), we obtain for low-frequency perturbations (τpω ≪ 1) the Korteweg-de Vries-Burgers (KdVB) equation:

$$\begin{split} \frac{\partial p}{\partial t} + c\_{\epsilon} \frac{\partial p}{\partial \mathbf{x}} + a\_{\epsilon} c\_{\epsilon}^{3} p \frac{\partial p}{\partial \mathbf{x}} - \beta\_{\epsilon} \frac{\partial^{2} p}{\partial \mathbf{x}^{2}} + \gamma\_{\epsilon} \frac{\partial^{3} p}{\partial \mathbf{x}^{3}} &= \mathbf{0}, \\ \beta\_{\epsilon} = \frac{c\_{\epsilon}^{2} \tau\_{p}}{2c\_{f}^{2}} \left( c\_{f}^{2} - c\_{\epsilon}^{2} \right), \quad \gamma\_{\epsilon} = \frac{c\_{\epsilon}^{3} \tau\_{p}^{2}}{8c\_{f}^{4}} \left( c\_{f}^{2} - c\_{\epsilon}^{2} \right) \left( c\_{f}^{2} - 5c\_{\epsilon}^{2} \right), \end{split} \tag{6}$$

whilst for high-frequency waves ðτpω ≫ 1Þ, we have obtained the following equation:

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

We restrict our attention to barotropic media. An equilibrium state equation of a barotropic medium is a one-parameter equation. As a result of relaxation, an addi-

and defines the completeness of the relaxation process. There are two limiting

i. Lack of relaxation (inner interaction processes are frozen) for which ξ ¼ 1:

ii. Relaxation which is complete (there is local thermodynamic equilibrium) for

p ¼ pð Þ ρ; ξ (1)

<sup>f</sup> ¼ dpf =dρ; (2)

<sup>e</sup> ¼ dpe=dρ: (3)

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup>: (4)

<sup>¼</sup> <sup>0</sup>: (5)

tional variable ξ (the inner parameter) appears in the state equation

<sup>p</sup> <sup>¼</sup> <sup>p</sup>ð Þ� <sup>ρ</sup>; <sup>1</sup> pfð Þ<sup>ρ</sup> , c<sup>2</sup>

<sup>p</sup> <sup>¼</sup> <sup>p</sup>ð Þ� <sup>ρ</sup>; <sup>0</sup> peð Þ<sup>ρ</sup> , c<sup>2</sup>

∂u

Slow and fast processes are compared by means of the relaxation time τp. To analyse the wave motion, we use the following hydrodynamic equations in

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup>, <sup>∂</sup><sup>u</sup>

∂t þ 1 ρ0

þ p � pe

The following dynamic state equation is applied to account for the relaxation effects:

Here <sup>V</sup> � <sup>ρ</sup>�<sup>1</sup> is the specific volume and <sup>x</sup> is the Lagrangian space coordinate. Clearly, for the fast processes ωτ<sup>p</sup> ≫ 1 , we have relation (2), and for the slow ones

The closed system of equations consists of two motion equations (4) and dynamic state equation (5). The motion equations (4) are written in Lagrangian coordinates since the state equation (5) is related to the element of mass of the medium.

The substantiation of (5) within the framework of the thermodynamics of irreversible processes has been given in [1, 2]. We note that the mechanisms of the exchange processes are not defined concretely when deriving the dynamic state equation (5). In this equation the thermodynamic and kinetic parameters appear only as sound velocities ce and cf and relaxation time τp. These are very common characteristics and they can be found experimentally. Hence, it is not necessary to

Combining the relationships (4) and (5), we obtain for low-frequency pertur-

∂2 p <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>e</sup>

> e τ2 p 8c<sup>4</sup> f c 2 <sup>f</sup> � c 2 e

, <sup>γ</sup><sup>e</sup> <sup>¼</sup> <sup>c</sup><sup>3</sup>

whilst for high-frequency waves ðτpω ≫ 1Þ, we have obtained the following

∂3 p <sup>∂</sup>x<sup>3</sup> <sup>¼</sup> <sup>0</sup>,

> c 2 <sup>f</sup> � 5c 2 e

,

(6)

bations (τpω ≪ 1) the Korteweg-de Vries-Burgers (KdVB) equation:

∂p

cases with corresponding sound velocities:

Research Advances in Chaos Theory

∂V <sup>∂</sup><sup>t</sup> � <sup>1</sup> ρ0

> τp dp dt � <sup>c</sup> 2 f dρ dt

know the inner exchange mechanism in detail.

∂p ∂t þ ce ∂p ∂x þ αec 3 ep ∂p <sup>∂</sup><sup>x</sup> � <sup>β</sup><sup>e</sup>

equation:

14

<sup>β</sup><sup>e</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup> e τp 2c<sup>2</sup> f c 2 <sup>f</sup> � c 2 e 

which ξ ¼ 0:

Lagrangian coordinates:

ωτ<sup>p</sup> ≪ 1 , we have (3).

$$\begin{split} & \frac{\partial^2 p}{\partial \mathbf{x}^2} - c\_f^{-2} \frac{\partial^2 p}{\partial t^2} + a\_f c\_f^2 \frac{\partial^2 p^2}{\partial \mathbf{x}^2} + \beta\_f \frac{\partial p}{\partial \mathbf{x}} + \gamma\_f p = \mathbf{0}, \\ & \beta\_f = \frac{c\_f^2 - c\_\epsilon^2}{\tau\_p c\_\epsilon^2 c\_f}, \quad \gamma\_f = \frac{c\_f^4 - c\_\epsilon^4}{2 \tau\_p^2 c\_\epsilon^4 c\_f^2}. \end{split} \tag{7}$$

Equation (6) with (β<sup>e</sup> ¼ 0) is the well-known the Korteweg-de Vries (KdV) equation. The investigation of the KdV equation in conjunction with the nonlinear Schrodinger (NLS) and sine-Gordon equations gives rise to the theory of solitons [4–13].

We focus our main attention on (7). It has a dissipative term β<sup>f</sup> ∂p=∂x and a dispersive term γ<sup>f</sup> p. Without the nonlinear and dissipative terms, we have a linear Klein-Gordon equation.

Let us write down (7) in dimensionless form. In the moving coordinate system with velocity cf , after factorization the equation has the form in the dimensionless variables

$$
\ddot{\mathbf{x}} = \sqrt{\frac{\eta\_f}{2}} (\mathbf{x} - c\_f t),
\ddot{\mathbf{t}} = \sqrt{\frac{\eta\_f}{2}} c\_f t,
\ddot{\mathbf{u}} = a\_f c\_f^2 p \text{ (tilde over variables } \ddot{\mathbf{x}}, \ddot{\mathbf{t}} \text{ and } \ddot{\mathbf{u}} \text{ is omitted)}
$$

$$
a\_f \mathbf{u} \cdot \mathbf{u} \quad \text{s.t.}
$$

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial}{\partial t} + u \frac{\partial}{\partial \mathbf{x}} \right) u + a \frac{\partial u}{\partial \mathbf{x}} + u = \mathbf{0}. \tag{8}$$

The constant <sup>α</sup> <sup>¼</sup> <sup>β</sup><sup>f</sup> <sup>=</sup> ffiffiffiffiffiffi 2γf p is always positive. Equation (8) without the dissipative term has the form of the nonlinear equation [14, 15]:

$$\frac{\partial}{\partial t}\left(\frac{\partial}{\partial t} + u\frac{\partial}{\partial \mathbf{x}}\right)u + u = 0. \tag{9}$$

Historically, (9) has been called the Vakhnenko equation (VE), and we will follow this name.

We note that (9) follows as a particular limit of the following generalized Korteweg-de Vries equation:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial \mathbf{x}} - \beta \frac{\partial^3 u}{\partial \mathbf{x}^3} \right) = \gamma u \tag{10}$$

derived by Ostrovsky [16] to model small-amplitude long waves in a rotating fluid (γu is induced by the Coriolis force) of finite depth. Subsequently, (9) was known by different names in the literature, such as the Ostrovsky-Hunter equation, the short-wave equation, the reduced Ostrovsky equation, and the Ostrovsky-Vakhnenko equation depending on the physical context in which it is studied.

The consideration here of (9) has interest from the viewpoint of the investigation of the propagation of high-frequency perturbations.

#### 2. Loop-like stationary solutions

The travelling wave solutions are solutions which are stationary with respect to a moving frame of reference. In this case, the evolution equation (a partial differential equation) becomes an ordinary differential equation (ODE) which is considerably easier to solve.

For the VE (9) it is convenient to introduce a new dependent variable z and new independent variables η and τ defined by

$$
\pi = (\mathfrak{u} - \mathfrak{v})/|\mathfrak{v}|, \quad \eta = (\mathfrak{x} - \mathfrak{v}t)/|\mathfrak{v}|^{1/2}, \quad \mathfrak{x} = t|\mathfrak{v}|^{1/2}, \tag{11}
$$

where v is a nonzero constant [15]. Then the VE becomes begin equation:

$$\left(z\_{\eta\tau} + \left(z z\_{\eta}\right)\_{\eta} + z + c = 0, \tag{12}$$

where c ¼ �1 corresponding to v≷0. We now seek stationary solutions of (12) for which z is a function of η only so that z<sup>τ</sup> ¼ 0 and z satisfies the ODE:

$$\left(z\mathbf{z}\_{\eta}\right)\_{\eta} + \mathbf{z} + \mathbf{c} = \mathbf{0}.\tag{13}$$

After one integration (13) gives

$$\begin{aligned} \frac{1}{2} \left( \mathbf{z} \mathbf{z}\_{\eta} \right)^{2} &= f(\mathbf{z}),\\ f(\mathbf{z}) &= -\frac{1}{3} \mathbf{z}^{3} - \frac{1}{2} \mathbf{c} \mathbf{z}^{2} + \frac{1}{6} \mathbf{A} = -\frac{1}{3} (\mathbf{z} - \mathbf{z}\_{1}) (\mathbf{z} - \mathbf{z}\_{2}) (\mathbf{z} - \mathbf{z}\_{3}). \end{aligned} \tag{14}$$

where A is a constant, and for periodic solutions z1, z<sup>2</sup> and z<sup>3</sup> are real constants such that z<sup>1</sup> ≤ z<sup>2</sup> ≤ z3. On using results 236.00 and 236.01 of [17], we may integrate (14) to obtain

$$\eta = \frac{\sqrt{\xi} \mathbf{z}\_1}{\sqrt{\mathbf{z}\_3 - \mathbf{z}\_1}} F(\boldsymbol{\rho}, \boldsymbol{m}) + \sqrt{\boldsymbol{\Phi}(\mathbf{z}\_3 - \mathbf{z}\_1)} E(\boldsymbol{\rho}, \boldsymbol{m}), \tag{15}$$

$$\sin \phi = \frac{z\_3 - z}{z\_3 - z\_2}, \quad m = \frac{z\_3 - z\_2}{z\_3 - z\_1}. \tag{16}$$

where Fð Þ φ; m and Eð Þ φ; m are incomplete elliptic integrals of the first and second kind, respectively. We have chosen the constant of integration in (15) to be zero so that z ¼ z<sup>3</sup> at η ¼ 0. The relations (15) give the required solution in parametric form, with z and η as functions of the parameter φ.

For c ¼ 1 (i.e., v> 0), there are periodic solutions for 0< A < 1 with λ<0, z<sup>2</sup> ∈ð Þ �1; 0 and z<sup>3</sup> ∈ð Þ 0; 0:5 ; an example of such a periodic wave is illustrated by curve 2 in Figure 1. Here we introduce a new independent variable ζ defined by

$$\frac{d\eta}{d\zeta} = \mathbf{z}.\tag{17}$$

solitary waves (18) are rather intriguing, it is the solution to the initial value problem that is of more interest in a physical context. An important question is the stability of the loop-like solutions. Although the analysis of stability does not link with the theory of solitons directly, the method applied in [15] is instructive, since it is successful in a nonlinear approximation. Stability of the loop-like solutions has been proved in [15]. From a physical viewpoint, the stability or otherwise of

The multivalued solutions obtained in Section 2 obviously mean that the study

of the VE (9) in the original coordinates ð Þ x; t leads to certain difficulties.

solutions is essential to their interpretation.

Figure 2.

17

Figure 1.

Loop-like Solitons

DOI: http://dx.doi.org/10.5772/intechopen.86583

Travelling wave solutions with v> 0.

Travelling wave solutions with v< 0.

3. The Vakhnenko-Parkes equation

A ¼ 1 gives the solitary wave limit:

$$
\mu = \frac{3}{2}\nu \operatorname{sech}^2(\zeta/2), \quad \eta = -\zeta + 3 \tanh\left(\zeta/2\right) \tag{18}
$$

as illustrated by curve 1 in Figure 1. The periodic waves and the solitary wave have a loop-like structure as illustrated in Figure 1. For c ¼ �1 (i.e., v< 0), there are periodic waves for �1< A <0 with λ>0, z<sup>2</sup> ∈ð Þ 0; 1 and z<sup>3</sup> ∈ð Þ 1; 1:5 ; an example of such a periodic wave is illustrated by curve 2 in Figure 2. When A ¼ 0 and λ ¼ 6, then the periodic wave solution simplifies to

$$u(\eta)/|\nu| = -\frac{1}{6}\eta^2 + \frac{1}{2}, \qquad -3 \le \eta \le 3, \qquad u(\eta + 6) = u(\eta). \tag{19}$$

This is shown by curve 1 in Figure 2. For A ≃ � 1 the solution has a sinusoidal form (curve 3 in Figure 2). Note that there are no solitary wave solutions.

A remarkable feature of the equation (9) is that it has a solitary wave (18) which has a loop-like form, i.e., it is a multivalued function (see Figure 1). Whilst loop

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

#### Figure 1.

where v is a nonzero constant [15]. Then the VE becomes begin equation:

where c ¼ �1 corresponding to v≷0. We now seek stationary solutions of (12) for which z is a function of η only so that z<sup>τ</sup> ¼ 0 and z satisfies the ODE:

where A is a constant, and for periodic solutions z1, z<sup>2</sup> and z<sup>3</sup> are real constants such that z<sup>1</sup> ≤ z<sup>2</sup> ≤ z3. On using results 236.00 and 236.01 of [17], we may integrate

<sup>F</sup>ð Þþ <sup>φ</sup>; <sup>m</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

6ð Þ z<sup>3</sup> � z<sup>1</sup>

z<sup>3</sup> � z<sup>1</sup>

, m <sup>¼</sup> <sup>z</sup><sup>3</sup> � <sup>z</sup><sup>2</sup>

<sup>η</sup> þ z þ c ¼ 0, (12)

<sup>η</sup> þ z þ c ¼ 0: (13)

(14)

ð Þ z � z<sup>1</sup> ð Þ z � z<sup>2</sup> ð Þ z � z<sup>3</sup> :

<sup>p</sup> <sup>E</sup>ð Þ <sup>φ</sup>; <sup>m</sup> , (15)

<sup>d</sup><sup>ζ</sup> <sup>¼</sup> <sup>z</sup>: (17)

ð Þ ζ=2 , η ¼ �ζ þ 3 tanh ð Þ ζ=2 (18)

, � 3≤η≤3, uð Þ¼ η þ 6 uð Þη : (19)

: (16)

zητ þ zz<sup>η</sup> � �

> zz<sup>η</sup> � �

After one integration (13) gives

� �<sup>2</sup> <sup>¼</sup> f zð Þ,

Research Advances in Chaos Theory

f zð Þ¼� <sup>1</sup>

3 <sup>z</sup><sup>3</sup> � <sup>1</sup> 2 cz<sup>2</sup> <sup>þ</sup> 1 <sup>6</sup> <sup>A</sup> ¼ � <sup>1</sup> 3

η ¼

A ¼ 1 gives the solitary wave limit:

<sup>u</sup> <sup>¼</sup> <sup>3</sup> 2 v sech2

then the periodic wave solution simplifies to

<sup>6</sup> <sup>η</sup><sup>2</sup> <sup>þ</sup> 1 2

<sup>u</sup>ð Þ<sup>η</sup> <sup>=</sup>∣v<sup>∣</sup> ¼ � <sup>1</sup>

16

ffiffiffi 6 <sup>p</sup> <sup>z</sup><sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi z<sup>3</sup> � z<sup>1</sup> p

sin <sup>φ</sup> <sup>¼</sup> <sup>z</sup><sup>3</sup> � <sup>z</sup>

parametric form, with z and η as functions of the parameter φ.

z<sup>3</sup> � z<sup>2</sup>

where Fð Þ φ; m and Eð Þ φ; m are incomplete elliptic integrals of the first and second kind, respectively. We have chosen the constant of integration in (15) to be zero so that z ¼ z<sup>3</sup> at η ¼ 0. The relations (15) give the required solution in

For c ¼ 1 (i.e., v> 0), there are periodic solutions for 0< A < 1 with λ<0, z<sup>2</sup> ∈ð Þ �1; 0 and z<sup>3</sup> ∈ð Þ 0; 0:5 ; an example of such a periodic wave is illustrated by curve 2 in Figure 1. Here we introduce a new independent variable ζ defined by

dη

as illustrated by curve 1 in Figure 1. The periodic waves and the solitary wave have a loop-like structure as illustrated in Figure 1. For c ¼ �1 (i.e., v< 0), there are periodic waves for �1< A <0 with λ>0, z<sup>2</sup> ∈ð Þ 0; 1 and z<sup>3</sup> ∈ð Þ 1; 1:5 ; an example of such a periodic wave is illustrated by curve 2 in Figure 2. When A ¼ 0 and λ ¼ 6,

This is shown by curve 1 in Figure 2. For A ≃ � 1 the solution has a sinusoidal

A remarkable feature of the equation (9) is that it has a solitary wave (18) which has a loop-like form, i.e., it is a multivalued function (see Figure 1). Whilst loop

form (curve 3 in Figure 2). Note that there are no solitary wave solutions.

1 <sup>2</sup> zz<sup>η</sup>

(14) to obtain

Travelling wave solutions with v> 0.

solitary waves (18) are rather intriguing, it is the solution to the initial value problem that is of more interest in a physical context. An important question is the stability of the loop-like solutions. Although the analysis of stability does not link with the theory of solitons directly, the method applied in [15] is instructive, since it is successful in a nonlinear approximation. Stability of the loop-like solutions has been proved in [15]. From a physical viewpoint, the stability or otherwise of solutions is essential to their interpretation.

#### 3. The Vakhnenko-Parkes equation

The multivalued solutions obtained in Section 2 obviously mean that the study of the VE (9) in the original coordinates ð Þ x; t leads to certain difficulties.

These difficulties can be avoided by writing down the VE in new independent coordinates. We have succeeded in finding these coordinates. Historically, working separately, we (Vyacheslav Vakhnenko in Ukraine and John Parkes in the UK) independently suggested such independent coordinates in which the solutions become one-valued functions. It is instructive to present the two derivations here. In one derivation a physical approach, namely, a transformation between Euler and Lagrange coordinates, was used, whereas in the other derivation, a pure mathematical approach was used.

Let us define new independent variables ð Þ X; T by the transformation

$$
\rho \, dT = d\mathbf{x} - \mathbf{u} \, dt, \qquad X = \mathbf{t}.\tag{20}
$$

WXXT þ ð Þ 1 þ WT WX ¼ 0: (27)

x ¼ θð Þ¼ X; T x<sup>0</sup> þ T þ W, t ¼ X, (29)

U X<sup>0</sup> ð Þ ; T dX<sup>0</sup> þ x1, t ¼ X, (30)

UT ¼ 0: (28)

: (31)

<sup>∣</sup>v<sup>∣</sup> (32)

U dζ � vζ: (33)

Z, where Z ≔ X � VT � X<sup>0</sup> (35)

: (34)

Alternatively, by eliminating φ between (22) and (24), we obtain

are given by

Loop-like Solitons

mathematical point of view.

terms of X and T.

so that

we obtain

19

new independent variables X and T defined by

DOI: http://dx.doi.org/10.5772/intechopen.86583

x ¼ T þ

ðX �∞

UUXXT � UXUXT <sup>þ</sup> <sup>U</sup><sup>2</sup>

Furthermore it follows from (21) that the original independent coordinates ð Þ x; t

where x<sup>0</sup> is an arbitrary constant. Since the functions θð Þ X; T and U Xð Þ ; T are single-valued, the problem of multivalued solutions has been resolved from the

Alternatively, in a pure mathematical approach, we may start by introducing

ðX �∞

Now, on introducing (25), (30) and (31) may be identified with (29) and (26),

The transformation into new coordinates, as has already been pointed out, was obtained by us independently of each other; nevertheless, we published the result together [18, 19]. Following the papers [20–23] hereafter, Eq. (27) (or in alternative

The travelling wave solution (15) and (16) for Equation (9) is also a travelling wave solution when written in terms of the transformed coordinates (X,T). In order to do this, we need to express the independent variable ζ, as introduced in (17), in

<sup>d</sup><sup>ζ</sup> <sup>¼</sup> <sup>U</sup> � <sup>v</sup>

ð

From the definition of η in (17), and the expressions for x and t given by (29),

½ � <sup>W</sup> � v Xð Þ � VT , where <sup>V</sup> <sup>≔</sup> <sup>v</sup>�<sup>1</sup>

UT dX<sup>0</sup>

where x<sup>1</sup> is an arbitrary constant. From (30), we obtain (23) but with

φð Þ¼ X; T 1 þ

respectively. The derivation of (27) and (28) proceeds as before.

form (28)) is referred to as the Vakhnenko-Parkes equation (VPE).

dη

∣v∣η ¼

From the expressions for z in (11) and (17), we obtain

∣v∣η ¼ j j v

1=2

ζ ¼ j j v 1=2

The expressions for ∣v∣η in (33) and (34) are equivalent if

The function φ is to be obtained. It is important that the functions x ¼ θð Þ X; T and u ¼ U Xð Þ ; T turn out to be single-valued. In terms of the coordinates ð Þ X; T , the solution of the VE (9) is given by single-valued parametric relations. The transformation into these coordinates is the key point in solving the problem of the interaction of solitons as well as explaining the multivalued solutions [3]. The transformation (20) is similar to the transformation between Eulerian coordinates ð Þ x; t and Lagrangian coordinates ð Þ X; T . We require that T ¼ x if there is no perturbation, i.e., if u xð Þ� ; t 0. Hence φ ¼ 1 when u xð Þ� ; t 0.

The function φ is the additional dependent variable in the equation system (22), (24) to which we reduce the original Eq. (9). We note that the transformation inverse to (20) is

$$d\boldsymbol{x} = \boldsymbol{\varphi} d\boldsymbol{T} + \boldsymbol{U} d\boldsymbol{X}, \quad \boldsymbol{t} = \boldsymbol{X}, \quad \boldsymbol{U}(\boldsymbol{X}, \boldsymbol{T}) \equiv \boldsymbol{u}(\boldsymbol{x}, \boldsymbol{t}).\tag{21}$$

It follows that

$$\frac{\partial \mathbf{x}}{\partial X} = U, \quad \frac{\partial \mathbf{x}}{\partial T} = \rho, \quad \frac{\partial t}{\partial X} = \mathbf{1}, \quad \frac{\partial t}{\partial T} = \mathbf{0}.$$

Hence

$$\frac{\partial \rho}{\partial X} = \frac{\partial U}{\partial T} \tag{22}$$

and

$$\frac{\partial}{\partial X} = \frac{\partial}{\partial t} + u \frac{\partial}{\partial \mathbf{x}}, \quad \frac{\partial}{\partial T} = \rho \frac{\partial}{\partial \mathbf{x}}.\tag{23}$$

By using (23), we can write Eq. (9) in terms of φð Þ X; T and U Xð Þ ; T , namely,

$$U\_{XT} + \varphi U = \mathbf{0}.\tag{24}$$

Equations (22) and (24) are the main system of equations. It can be reduced to a nonlinear equation (27) in one unknown W defined by

$$W\_X = \mathcal{U}.\tag{25}$$

From (22), (25) and the requirement that φ ¼ 1 when U � 0, we have

$$
\rho = \mathbf{1} + \mathbf{W}\_T.\tag{26}
$$

Then, by eliminating φ and U between (24), (25) and (26), we arrive at a transformed form of the VE (9), namely,

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

These difficulties can be avoided by writing down the VE in new independent coordinates. We have succeeded in finding these coordinates. Historically, working separately, we (Vyacheslav Vakhnenko in Ukraine and John Parkes in the UK) independently suggested such independent coordinates in which the solutions become one-valued functions. It is instructive to present the two derivations here. In one derivation a physical approach, namely, a transformation between Euler and Lagrange coordinates, was used, whereas in the other derivation, a pure mathematical approach was used. Let us define new independent variables ð Þ X; T by the transformation

The function φ is to be obtained. It is important that the functions x ¼ θð Þ X; T and u ¼ U Xð Þ ; T turn out to be single-valued. In terms of the coordinates ð Þ X; T , the solution of the VE (9) is given by single-valued parametric relations. The transformation into these coordinates is the key point in solving the problem of the interaction of solitons as well as explaining the multivalued solutions [3]. The transformation (20) is similar to the transformation between Eulerian coordinates ð Þ x; t and Lagrangian coordinates ð Þ X; T . We require that T ¼ x if there is no

The function φ is the additional dependent variable in the equation system (22),

dx ¼ φdT þ U dX, t ¼ X, U Xð Þ� ; T u xð Þ ; t : (21)

<sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>1</sup>, <sup>∂</sup><sup>t</sup>

<sup>∂</sup><sup>T</sup> <sup>¼</sup> <sup>φ</sup>

∂

UXT þ φU ¼ 0: (24)

WX ¼ U: (25)

φ ¼ 1 þ WT: (26)

<sup>∂</sup><sup>T</sup> <sup>¼</sup> <sup>0</sup>:

<sup>∂</sup><sup>T</sup> (22)

<sup>∂</sup><sup>x</sup> : (23)

(24) to which we reduce the original Eq. (9). We note that the transformation

<sup>∂</sup><sup>T</sup> <sup>¼</sup> <sup>φ</sup>, <sup>∂</sup><sup>t</sup>

∂φ <sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>∂</sup><sup>U</sup>

By using (23), we can write Eq. (9) in terms of φð Þ X; T and U Xð Þ ; T , namely,

Equations (22) and (24) are the main system of equations. It can be reduced to a

From (22), (25) and the requirement that φ ¼ 1 when U � 0, we have

Then, by eliminating φ and U between (24), (25) and (26), we arrive at a

perturbation, i.e., if u xð Þ� ; t 0. Hence φ ¼ 1 when u xð Þ� ; t 0.

∂x

<sup>∂</sup><sup>X</sup> <sup>¼</sup> U, <sup>∂</sup><sup>x</sup>

∂ <sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>∂</sup> ∂t þ u ∂ <sup>∂</sup><sup>x</sup> , <sup>∂</sup>

nonlinear equation (27) in one unknown W defined by

transformed form of the VE (9), namely,

inverse to (20) is

Research Advances in Chaos Theory

It follows that

Hence

and

18

φdT ¼ dx � udt, X ¼ t: (20)

$$(\mathbf{W}\_{\rm XXT} + (\mathbf{1} + \mathbf{W}\_T)\mathbf{W}\_X = \mathbf{0}.\tag{27}$$

Alternatively, by eliminating φ between (22) and (24), we obtain

$$
\mathbf{U}\mathbf{U}\_{\rm XXT} - \mathbf{U}\_{\rm X}\mathbf{U}\_{\rm XT} + \mathbf{U}^2\mathbf{U}\_T = \mathbf{0}.\tag{28}
$$

Furthermore it follows from (21) that the original independent coordinates ð Þ x; t are given by

$$\mathbf{x} = \theta(\mathbf{X}, T) = \mathbf{x}\_0 + T + \mathcal{W}, \qquad t = \mathbf{X}, \tag{29}$$

where x<sup>0</sup> is an arbitrary constant. Since the functions θð Þ X; T and U Xð Þ ; T are single-valued, the problem of multivalued solutions has been resolved from the mathematical point of view.

Alternatively, in a pure mathematical approach, we may start by introducing new independent variables X and T defined by

$$\boldsymbol{\omega} = \boldsymbol{T} + \int\_{-\infty}^{\boldsymbol{X}} \boldsymbol{U}(\boldsymbol{X}', \boldsymbol{T}) \boldsymbol{d} \boldsymbol{X}' + \boldsymbol{\varkappa}\_{\rm 1} \qquad \boldsymbol{t} = \boldsymbol{X}, \tag{30}$$

where x<sup>1</sup> is an arbitrary constant. From (30), we obtain (23) but with

$$\varrho(\mathbf{X}, T) = \mathbf{1} + \int\_{-\infty}^{\mathbf{X}} U\_T d\mathbf{X}'.\tag{31}$$

Now, on introducing (25), (30) and (31) may be identified with (29) and (26), respectively. The derivation of (27) and (28) proceeds as before.

The transformation into new coordinates, as has already been pointed out, was obtained by us independently of each other; nevertheless, we published the result together [18, 19]. Following the papers [20–23] hereafter, Eq. (27) (or in alternative form (28)) is referred to as the Vakhnenko-Parkes equation (VPE).

The travelling wave solution (15) and (16) for Equation (9) is also a travelling wave solution when written in terms of the transformed coordinates (X,T). In order to do this, we need to express the independent variable ζ, as introduced in (17), in terms of X and T.

From the expressions for z in (11) and (17), we obtain

$$\frac{d\eta}{d\zeta} = \frac{U - v}{|v|}\tag{32}$$

so that

$$|\nu|\eta = \int U d\zeta - \nu \zeta. \tag{33}$$

From the definition of η in (17), and the expressions for x and t given by (29), we obtain

$$|\boldsymbol{\upsilon}|\boldsymbol{\eta} = |\boldsymbol{\upsilon}|^{1/2} [\boldsymbol{W} - \boldsymbol{\upsilon}(\boldsymbol{X} - \boldsymbol{V}\boldsymbol{T})], \quad \text{where} \quad \boldsymbol{V} \coloneqq \boldsymbol{\upsilon}^{-1}. \tag{34}$$

The expressions for ∣v∣η in (33) and (34) are equivalent if

$$\mathcal{L} = |\boldsymbol{\nu}|^{1/2}\mathcal{Z}, \quad \text{where} \quad \mathcal{Z} \coloneqq \mathcal{X} - \mathcal{V}\mathcal{T} - \mathcal{X}\_0 \tag{35}$$

and X<sup>0</sup> is an arbitrary constant, so that

$$W = \int U dZ \text{ and } U = W\_Z. \tag{36}$$

Hence, from (34), it follows that

$$\mathcal{W} = \frac{\sqrt{|\boldsymbol{v}|}}{p} [(\mathbf{z}\_1 + \boldsymbol{c})\boldsymbol{w} + (\mathbf{z}\_3 - \mathbf{z}\_1)E(\boldsymbol{w}|\boldsymbol{m})] + \mathcal{W}\_0. \tag{37}$$

where <sup>w</sup> <sup>¼</sup> <sup>p</sup> ffiffiffiffiffi <sup>∣</sup>v<sup>∣</sup> <sup>p</sup> <sup>Z</sup> and <sup>W</sup><sup>0</sup> is an arbitrary constant. Then

$$\frac{U}{|v|} = c + z\_3 - (z\_3 - z\_2) \text{sn}^2(w|m), \quad \text{where } w = p\sqrt{|v|}Z. \tag{38}$$

Eqs. (37) and (38) give the travelling wave solutions to the VPE in the forms (27) and (28), respectively. Eq. (38) is also the travelling wave solution of the VE (9) expressed in terms of the new coordinates (X,T). In the limiting case m ¼ 1, (38) gives a solitary wave in the following two forms: For v> 0

$$U/v = \frac{3}{2}\text{sech}^2\left(\frac{1}{2}\sqrt{v}Z\right) \tag{39}$$

There are two important observations to be made. Firstly, all the travelling wave solutions in terms of the new coordinates are single-valued. Secondly, the periodic solution shown by curve 1 in Figure 2, i.e., the solution consisting of parabolas, is not periodic in terms of the new coordinates. Hence, we reveal some accordance between curve 1 in Figure 3 and curve 1 in Figure 4. These features are important

The Hirota method gives the N-soliton solution as well as enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering method [24]. Thus, the Hirota method allows us to formulate the inverse scattering method which is the most appropriate way of tackling the initial value problem (Cauchy problem).

In the Hirota method the equation, in our case the VPE (27), under investigation

<sup>f</sup> � <sup>f</sup> <sup>¼</sup> <sup>0</sup>, (41)

a Tð Þ ;X b T<sup>0</sup>

W ¼ 6 lnð Þf <sup>X</sup>, (42)

;X<sup>0</sup> ð Þ T¼T<sup>0</sup> ,X¼X<sup>0</sup>

: (43)

for finding the solutions by the inverse scattering method [24–30].

Travelling wave solutions with v< 0 in coordinates (X,T).

4. From Hirota method to the inverse scattering method

should be transformed into the Hirota bilinear form [9, 24]:

with

Figure 4.

Loop-like Solitons

DOI: http://dx.doi.org/10.5772/intechopen.86583

Dn TD<sup>m</sup>

21

<sup>X</sup> <sup>a</sup> � <sup>b</sup> <sup>¼</sup> <sup>∂</sup>

<sup>∂</sup><sup>T</sup> � <sup>∂</sup> ∂T<sup>0</sup> <sup>n</sup> ∂

DTD<sup>3</sup>

<sup>X</sup> <sup>þ</sup> <sup>D</sup><sup>2</sup> X

The Hirota bilinear D-operator is defined as (see Section 5.2 in [9])

<sup>∂</sup><sup>X</sup> � <sup>∂</sup> ∂X<sup>0</sup> <sup>m</sup>

Now we present a Bäcklund transformation for VPE (27) written in the bilinear form (41). This type of Bäcklund transformation was first introduced by Hirota [31]

and, for v< 0,

$$U/|\nu| = -\mathbf{1} + \frac{3}{2}\text{sech}^2\left(\frac{\mathbf{1}}{2}\sqrt{|\nu|}Z\right). \tag{40}$$

These two solutions are illustrated by curve 1 in Figures 3 and 4, respectively. The other curves illustrate examples of the solution given by (38) when m 6¼ 1. Curves 1 and 2 in Figure 3 relate to curves 1 and 2, respectively, in Figure 1. Curves 1, 2 and 3 in Figure 4 relate to curves 1, 2 and 3, respectively, in Figure 2.

Figure 3. Travelling wave solutions with v>0 in coordinates (X,T).

and X<sup>0</sup> is an arbitrary constant, so that

Hence, from (34), it follows that

Research Advances in Chaos Theory

W ¼

where <sup>w</sup> <sup>¼</sup> <sup>p</sup> ffiffiffiffiffi

and, for v< 0,

Figure 3.

20

Travelling wave solutions with v>0 in coordinates (X,T).

U ∣v∣ W ¼ ð

<sup>∣</sup>v<sup>∣</sup> <sup>p</sup> <sup>Z</sup> and <sup>W</sup><sup>0</sup> is an arbitrary constant. Then

Eqs. (37) and (38) give the travelling wave solutions to the VPE in the forms (27) and (28), respectively. Eq. (38) is also the travelling wave solution of the VE (9) expressed in terms of the new coordinates (X,T). In the limiting case m ¼ 1,

> sech<sup>2</sup> <sup>1</sup> 2 ffiffi <sup>v</sup> <sup>p</sup> <sup>Z</sup> � �

> > sech<sup>2</sup> <sup>1</sup> 2

3 2

1, 2 and 3 in Figure 4 relate to curves 1, 2 and 3, respectively, in Figure 2.

These two solutions are illustrated by curve 1 in Figures 3 and 4, respectively. The other curves illustrate examples of the solution given by (38) when m 6¼ 1. Curves 1 and 2 in Figure 3 relate to curves 1 and 2, respectively, in Figure 1. Curves

ffiffiffiffiffi <sup>∣</sup>v<sup>∣</sup> <sup>p</sup>

<sup>¼</sup> <sup>c</sup> <sup>þ</sup> <sup>z</sup><sup>3</sup> � ð Þ <sup>z</sup><sup>3</sup> � <sup>z</sup><sup>2</sup> sn<sup>2</sup>

(38) gives a solitary wave in the following two forms: For v> 0

<sup>U</sup>=<sup>v</sup> <sup>¼</sup> <sup>3</sup> 2

U=∣v∣ ¼ �1 þ

U dZ and U ¼ WZ: (36)

<sup>p</sup> ½ð Þ <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup> <sup>w</sup> <sup>þ</sup> ð Þ <sup>z</sup><sup>3</sup> � <sup>z</sup><sup>1</sup> E wð Þ <sup>j</sup><sup>m</sup> � þ <sup>W</sup>0, (37)

ð Þ <sup>w</sup>j<sup>m</sup> , where <sup>w</sup> <sup>¼</sup> <sup>p</sup> ffiffiffiffiffi

ffiffiffiffiffi ∣v∣ p Z � � ∣v∣ p Z: (38)

: (40)

(39)

Figure 4. Travelling wave solutions with v< 0 in coordinates (X,T).

There are two important observations to be made. Firstly, all the travelling wave solutions in terms of the new coordinates are single-valued. Secondly, the periodic solution shown by curve 1 in Figure 2, i.e., the solution consisting of parabolas, is not periodic in terms of the new coordinates. Hence, we reveal some accordance between curve 1 in Figure 3 and curve 1 in Figure 4. These features are important for finding the solutions by the inverse scattering method [24–30].

#### 4. From Hirota method to the inverse scattering method

The Hirota method gives the N-soliton solution as well as enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering method [24]. Thus, the Hirota method allows us to formulate the inverse scattering method which is the most appropriate way of tackling the initial value problem (Cauchy problem).

In the Hirota method the equation, in our case the VPE (27), under investigation should be transformed into the Hirota bilinear form [9, 24]:

$$(D\_T D\_X^3 + D\_X^2) f \cdot f = 0,\tag{41}$$

with

$$\mathcal{W} = \mathsf{G}(\ln f)\_{\mathcal{X}^\flat} \tag{42}$$

The Hirota bilinear D-operator is defined as (see Section 5.2 in [9])

$$D\_T^\pi D\_X^m a \cdot b = \left(\frac{\partial}{\partial T} - \frac{\partial}{\partial T'}\right)^n \left(\frac{\partial}{\partial X} - \frac{\partial}{\partial X'}\right)^m a(T, X)b(T', X')\Big|\_{T = T', X = X'}.\tag{43}$$

Now we present a Bäcklund transformation for VPE (27) written in the bilinear form (41). This type of Bäcklund transformation was first introduced by Hirota [31] and has the advantage that the transformation equations are linear with respect to each dependent variable. This Bäcklund transformation can be transformed to the ordinary one [24]:

$$(D\_X^3 - \lambda)f' \cdot f = \mathbf{0},\tag{44}$$

Now, according to (62) and (72), the inverse scattering method restricts the solutions to those that vanish as ∣X∣ ! ∞, so h Tð Þ is to be identically zero. Thus, the pair of equations (47) and (48) or (47) and (50) can be considered as the Lax pair for the

Since (47) and (48) are alternative forms of Eqs. (44) and (45), respectively, it follows that the pair of equations (47) and (48) is associated with the VPE (27) considered here. Thus, the IST problem is directly related to a spectral equation of third order, namely, (47). The inverse problem for certain third-order spectral equations has been considered by Kaup [33] and Caudrey [34, 35]. As expected, (47) and (48) are similar to, but cannot be transformed into, the corresponding equations for the Hirota-Satsuma equation (HSE) (see Eq. (A8a) and (A8b) in [39]). Clarkson and Mansfield [40] note that the scattering problem for the HSE is similar to that for the Boussinesq equation which has been studied comprehensively

5. The inverse scattering method for a third-order equation

k ¼

W Xð Þ¼ ; <sup>0</sup> <sup>6</sup> ffiffiffi

is the initial condition for the VPE.

<sup>ψ</sup>ð Þ <sup>X</sup>; <sup>0</sup>; <sup>ζ</sup> exp ð Þ¼ �ζ<sup>X</sup> <sup>1</sup> � <sup>β</sup><sup>1</sup> exp ffiffiffi

Eq. (48), we may assume that

23

ffiffiffi 3 p

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup>

<sup>2</sup> <sup>ξ</sup>1, <sup>α</sup> <sup>¼</sup> <sup>1</sup>

∂ ∂X

5.1 Example of the use of the IST method to find the one-soliton solution

Consider the one-soliton solution of the VPE by application of the IST method.

For convenience we introduce new notation ξ<sup>1</sup> and β<sup>1</sup> instead of parameters k

2

ln 1 <sup>þ</sup> <sup>β</sup><sup>1</sup> 2 ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup>

The first step in the IST method is to solve the spectral equation (47) with spectral parameter λ for the given initial condition W Xð Þ ; 0 . In our example it is (54). The solution is studied over the complex <sup>ζ</sup>-plane, where <sup>ζ</sup><sup>3</sup> <sup>¼</sup> <sup>λ</sup>. One can verify by direct substitution of (55) in (47) that the solution ψð Þ X; 0; ζ of the linear ODE

<sup>3</sup> <sup>p</sup> <sup>ξ</sup>1<sup>X</sup> � �

where <sup>ω</sup><sup>j</sup> <sup>¼</sup> <sup>e</sup>i2πð Þ <sup>j</sup>�<sup>1</sup> <sup>=</sup><sup>3</sup> are the cube roots of 1 (<sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3). The constants <sup>β</sup><sup>1</sup> and <sup>ξ</sup>1,

exp ffiffi 3 p ð Þ <sup>ξ</sup>1<sup>X</sup> <sup>2</sup> ffiffi 3 <sup>p</sup> <sup>ξ</sup><sup>1</sup>

The second step in the IST method is to obtain the evolution of β<sup>1</sup> and ξ1. The time dependence of the solution ψð Þ X; T is described by Eq. (48). Analysing

(47), normalized so that ψð Þ X; 0; ζ exp ð Þ! �ζX 1 at X ! �∞, is given by

1 þ β<sup>1</sup>

as we will show, are associated with the local spectral data.

W Xð Þ¼ ; 0 6kð Þ 1 þ tanh ð Þη , η ¼ kX þ α: (52)

ln <sup>β</sup>1=<sup>2</sup> ffiffiffi

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup>

exp ffiffiffi 3 <sup>p</sup> <sup>ξ</sup>1<sup>X</sup> � � � � (54)

ω2

<sup>i</sup>ω2ξ<sup>1</sup> � <sup>ζ</sup> <sup>þ</sup> <sup>ω</sup><sup>3</sup>

�iω3ξ<sup>1</sup> � ζ � �, (55)

� � (53)

VPE (27).

Loop-like Solitons

DOI: http://dx.doi.org/10.5772/intechopen.86583

by Deift et al. [37].

and α by

then

Let the initial perturbation be

$$(\mathbf{3D}\_X \mathbf{D}\_T + \mathbf{1} + \mu \mathbf{D}\_X) \mathbf{f}' \cdot \mathbf{f} = \mathbf{0},\tag{45}$$

where λ ¼ λð Þ X is an arbitrary function of X and μ ¼ μð Þ T is an arbitrary function of T.

The inverse scattering transform (IST) method is arguably the most important discovery in the theory of solitons. The method enables one to solve the initial value problem for a nonlinear evolution equation. Moreover, it provides a proof of the complete integrability of the equation.

The essence of the application of the IST is as follows. The initial equation VPE (27) is written as the compatibility condition for two linear equations. These equations are presented in (47) and (48). Then W Xð Þ ; 0 is mapped into the scattering Sð Þ 0 for (47). It is important that since the variable W Xð Þ ; T contained in the spectral equation (47) evolves according to (27), the spectrum λ always retains constant values. The time evolution of S Tð Þ is simple and linear. From a knowledge of S Tð Þ, we reconstruct W Xð Þ ; T .

The use of the IST is the most appropriate way of tackling the initial value problem. In order to apply the IST method, one first has to formulate the associated eigenvalue problem. This can be achieved by finding a Bäcklund transformation associated with the VPE.

Now we will show that the IST problem for the VPE in the form (27) has a thirdorder eigenvalue problem that is similar to the one associated with a higher-order KdV equation [32, 33], a Boussinesq equation [33–37] and a model equation for shallow water waves [9, 38].

Introducing the function

$$
\psi = f'/f,\tag{46}
$$

and taking into account (42), we find that (44) and (45) reduce to

$$
\lambda \boldsymbol{\mu}\_{XXX} + \boldsymbol{\mathcal{W}}\_{X} \boldsymbol{\upmu}\_{X} - \lambda \boldsymbol{\upmu} = \mathbf{0},
\tag{47}
$$

$$
\mathfrak{B}\mu\_{XT} + (\mathfrak{1} + W\_T)\mathfrak{y} + \mu\mathfrak{y}\_X = \mathbf{0},
\tag{48}
$$

respectively, where we have used results similar to (X.1)–(X.3) in [9]. From (47) and (48), it can be shown that

$$\left[3\lambda\upmu\_{T} + (\mathbb{1} + \mathcal{W}\_{T})\upmu\_{XX} - \mathcal{W}\_{XT}\upmu\_{X} + [\mathcal{W}\_{X\mathcal{X}} + (\mathbb{1} + \mathcal{W}\_{T})\mathcal{W}\_{X} + \mu\mathbb{1}]\upmu = \mathbf{0} \tag{49}$$

and

$$[\mathcal{W}\_{\rm XXT} + (\mathbf{1} + \mathcal{W}\_T)\mathcal{W}\_X]\_X \boldsymbol{\psi} + (\mathbf{3}\boldsymbol{\psi}\_T + \boldsymbol{\mu}\boldsymbol{\psi})\boldsymbol{\lambda}\_X = \mathbf{0}.\tag{50}$$

In view of (27), (49) becomes

$$
\Im \lambda \boldsymbol{\mu}\_T + (\mathbf{1} + \mathbf{W}\_T) \boldsymbol{\mu}\_{XX} - \mathbf{W}\_{XT} \boldsymbol{\mu}\_X + \lambda \boldsymbol{\mu} \boldsymbol{\mu} = \mathbf{0},\tag{51}
$$

and (50) implies that λ<sup>X</sup> ¼ 0 so the spectrum λ of (47) remains constant. Constant λ is what is required in the IST problem. Equation (50) yields the equation WXXT þ ð Þ 1 þ WT WX ¼ h Tð Þ, where h Tð Þ is an arbitrary function of T.

and has the advantage that the transformation equations are linear with respect to each dependent variable. This Bäcklund transformation can be transformed

0

The inverse scattering transform (IST) method is arguably the most important discovery in the theory of solitons. The method enables one to solve the initial value problem for a nonlinear evolution equation. Moreover, it provides a proof of the

The essence of the application of the IST is as follows. The initial equation VPE (27) is written as the compatibility condition for two linear equations. These equations are presented in (47) and (48). Then W Xð Þ ; 0 is mapped into the scattering Sð Þ 0 for (47). It is important that since the variable W Xð Þ ; T contained in the spectral equation (47) evolves according to (27), the spectrum λ always retains constant values. The time evolution of S Tð Þ is simple and linear. From a knowledge

The use of the IST is the most appropriate way of tackling the initial value problem. In order to apply the IST method, one first has to formulate the associated eigenvalue problem. This can be achieved by finding a Bäcklund transformation

> ψ ¼ f 0

and taking into account (42), we find that (44) and (45) reduce to

respectively, where we have used results similar to (X.1)–(X.3) in [9].

3λψ<sup>T</sup> þ ð Þ 1 þ WT ψXX � WXTψ<sup>X</sup> þ ½ � WXXT þ ð Þ 1 þ WT WX þ μλ ψ ¼ 0 (49)

and (50) implies that λ<sup>X</sup> ¼ 0 so the spectrum λ of (47) remains constant. Constant λ is what is required in the IST problem. Equation (50) yields the equation

WXXT þ ð Þ 1 þ WT WX ¼ h Tð Þ, where h Tð Þ is an arbitrary function of T.

½ � WXXT þ ð Þ 1 þ WT WX <sup>X</sup>ψ þ ð Þ 3ψ<sup>T</sup> þ μψ λ<sup>X</sup> ¼ 0: (50)

3λψ<sup>T</sup> þ ð Þ 1 þ WT ψXX � WXTψ<sup>X</sup> þ λμψ ¼ 0, (51)

From (47) and (48), it can be shown that

In view of (27), (49) becomes

Now we will show that the IST problem for the VPE in the form (27) has a thirdorder eigenvalue problem that is similar to the one associated with a higher-order KdV equation [32, 33], a Boussinesq equation [33–37] and a model equation for

0

� f ¼ 0, (44)

=f, (46)

ψXXX þ WXψ<sup>X</sup> � λψ ¼ 0, (47) 3ψXT þ ð Þ 1 þ WT ψ þ μψ<sup>X</sup> ¼ 0, (48)

� f ¼ 0, (45)

D3 <sup>X</sup> � <sup>λ</sup> <sup>f</sup>

ð Þ 3DXDT þ 1 þ μDX f

where λ ¼ λð Þ X is an arbitrary function of X and μ ¼ μð Þ T is an arbitrary

to the ordinary one [24]:

Research Advances in Chaos Theory

complete integrability of the equation.

of S Tð Þ, we reconstruct W Xð Þ ; T .

associated with the VPE.

shallow water waves [9, 38]. Introducing the function

and

22

function of T.

Now, according to (62) and (72), the inverse scattering method restricts the solutions to those that vanish as ∣X∣ ! ∞, so h Tð Þ is to be identically zero. Thus, the pair of equations (47) and (48) or (47) and (50) can be considered as the Lax pair for the VPE (27).

Since (47) and (48) are alternative forms of Eqs. (44) and (45), respectively, it follows that the pair of equations (47) and (48) is associated with the VPE (27) considered here. Thus, the IST problem is directly related to a spectral equation of third order, namely, (47). The inverse problem for certain third-order spectral equations has been considered by Kaup [33] and Caudrey [34, 35]. As expected, (47) and (48) are similar to, but cannot be transformed into, the corresponding equations for the Hirota-Satsuma equation (HSE) (see Eq. (A8a) and (A8b) in [39]). Clarkson and Mansfield [40] note that the scattering problem for the HSE is similar to that for the Boussinesq equation which has been studied comprehensively by Deift et al. [37].

#### 5. The inverse scattering method for a third-order equation

#### 5.1 Example of the use of the IST method to find the one-soliton solution

Consider the one-soliton solution of the VPE by application of the IST method. Let the initial perturbation be

$$W(X,0) = \mathfrak{G}k(\mathfrak{1} + \tanh(\eta)), \quad \eta = kX + a. \tag{52}$$

For convenience we introduce new notation ξ<sup>1</sup> and β<sup>1</sup> instead of parameters k and α by

$$k = \frac{\sqrt{3}}{2}\xi\_1, \quad a = \frac{1}{2}\ln\left(\beta\_1/2\sqrt{3}\xi\_1\right) \tag{53}$$

then

$$\mathcal{W}(\mathbf{X}, \mathbf{0}) = 6\sqrt{3}\xi\_1 \frac{\partial}{\partial \mathbf{X}} \ln\left[\mathbf{1} + \frac{\beta\_1}{2\sqrt{3}\xi\_1} \exp\left(\sqrt{3}\xi\_1 \mathbf{X}\right)\right] \tag{54}$$

is the initial condition for the VPE.

The first step in the IST method is to solve the spectral equation (47) with spectral parameter λ for the given initial condition W Xð Þ ; 0 . In our example it is (54). The solution is studied over the complex <sup>ζ</sup>-plane, where <sup>ζ</sup><sup>3</sup> <sup>¼</sup> <sup>λ</sup>. One can verify by direct substitution of (55) in (47) that the solution ψð Þ X; 0; ζ of the linear ODE (47), normalized so that ψð Þ X; 0; ζ exp ð Þ! �ζX 1 at X ! �∞, is given by

$$\Psi(X,0;\zeta)\exp\left(-\zeta X\right) = 1 - \frac{\beta\_1 \exp\left(\sqrt{3}\xi\_1 X\right)}{1 + \beta\_1 \frac{\exp\left(\sqrt{3}\xi\_1 X\right)}{2\sqrt{3}\xi\_1}} \left[\frac{a\nu\_2}{\mathrm{i}a\nu\_2\xi\_1 - \zeta} + \frac{a\nu\_3}{-\mathrm{i}a\nu\_3\xi\_1 - \zeta}\right],\tag{55}$$

where <sup>ω</sup><sup>j</sup> <sup>¼</sup> <sup>e</sup>i2πð Þ <sup>j</sup>�<sup>1</sup> <sup>=</sup><sup>3</sup> are the cube roots of 1 (<sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3). The constants <sup>β</sup><sup>1</sup> and <sup>ξ</sup>1, as we will show, are associated with the local spectral data.

The second step in the IST method is to obtain the evolution of β<sup>1</sup> and ξ1. The time dependence of the solution ψð Þ X; T is described by Eq. (48). Analysing Eq. (48), we may assume that

$$\begin{aligned} \xi\_1(T) &= \xi\_1(0) = \text{const.},\\ \beta\_1(T) &= \beta\_1(0) \exp\left(-\frac{1}{\sqrt{3}\xi\_1}T\right). \end{aligned} \tag{56}$$

where, as previously, <sup>ω</sup><sup>j</sup> <sup>¼</sup> <sup>e</sup>2πið Þ <sup>j</sup>�<sup>1</sup> <sup>=</sup><sup>3</sup> are the cube roots of 1 (<sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3). Obvi-

The solution of the linear equation (47) (or equivalently (59)) has been obtained

Caudrey [34] showed how Eq. (59) can be solved by expressing it as a Fredholm

The information about the singularities of the Jost functions ϕjð Þ X; ζ reside in the

n j ¼ 1 j 6¼ i

and it can be found because we know the solution (47) in any regular regions

Now we consider the singularities on the boundaries between regions. However, in order to simplify matters, we first make some observations. The solution of the spectral problem can be facilitated by using various symmetry properties. In view of

0

B@

means we need only to consider ϕ1ð Þ X; ζ . In our case, for ϕ1ð Þ X; ζ , the complex

ϕið Þ X; ζ ϕið Þ X; ζ <sup>X</sup> ϕið Þ X; ζ XX 1

ϕ1ð Þ¼ X; ζ=ω<sup>1</sup> ϕ2ð Þ¼ X; ζ=ω<sup>2</sup> ϕ3ð Þ X; ζ=ω<sup>3</sup> (65)

γ ð Þk ij ϕ<sup>j</sup> X; ζ

spectral data. First let us consider the poles. It is assumed that a pole ζ

boundary between two regions. Then, as proven in [34], the residue is

ð Þk i � � <sup>¼</sup> <sup>∑</sup>

from solving the direct problem (see Section 5.2). Note that, for ϕ<sup>j</sup> X; ζ

<sup>i</sup> lies in the interior of a regular region. The quantities ζ

ϕið Þ¼ X; ζ

ζ-plane is divided into four regions by two lines (see Figure 5) given by

Res ϕ<sup>i</sup> X; ζ

The complex ζ-plane is to be divided into regions such that, in the interior of each region, the order of the numbers Reð Þ λið Þζ is fixed. As we pass from one region to another, this order changes, and hence, on a boundary between two regions, Reð Þ¼ <sup>λ</sup>ið Þ<sup>ζ</sup> Re <sup>λ</sup>jð Þ<sup>ζ</sup> � � for at least one pair <sup>i</sup> 6¼ <sup>j</sup>. The Jost function <sup>ϕ</sup><sup>j</sup> is regular throughout the complex ζ-plane apart from poles and finite singularities on the boundaries between the regions. At any point in the interior of any region of the complex ζ-plane, the solution of Eq. (59) is obtained by the relation (2.12) from

<sup>Φ</sup>jð Þ <sup>X</sup>; <sup>ζ</sup> <sup>≔</sup> exp �λjð Þ<sup>ζ</sup> <sup>X</sup> � �ϕjð Þ! <sup>X</sup>; <sup>ζ</sup> vjð Þ<sup>ζ</sup> as <sup>X</sup> ! �∞: (62)

ð Þζ and vjð Þζ are analytic throughout the

ð Þ X; ζ which have the asymptotic

ð Þk

ð Þk i � �, the

ij constitute

ð Þk

ð Þ X; ζ and j 6¼ i and does not lie on a

� � (63)

ð Þk <sup>i</sup> and γ

CA, (64)

ð Þk i

<sup>i</sup> in ϕið Þ X; ζ

ously the λjð Þζ are distinct, and they and v~<sup>j</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86583

by Caudrey [34] in terms of Jost functions ϕ<sup>j</sup>

[34]. It is the direct spectral problem.

is simple, does not coincide with a pole of ϕ<sup>j</sup>

the discrete part of the spectral data.

whilst the symmetry

(47), we need only consider the first elements of

complex ζ-plane.

Loop-like Solitons

integral equation.

5.3 The spectral data

point ζ ð Þk

25

behaviour:

Below, the assumption of these relationships will be justified. Indeed, we know that the spectrum λ in (47) remains constant if W Xð Þ ; T evolves according to Eq. (27). Therefore, as will be proved, the spectrum data evolve as in (70). In notations (77) and (78), from (70) we obtain the relations (56).

The final step in IST method is to select the solution W Xð Þ ; T from (55) with ξ1ð Þ T , β1ð Þ T as in (56). According to Eq. (2.7) in [33], we expand ψð Þ X; T; ζ as an asymptotic series in ζ�<sup>1</sup> to obtain

$$\Psi(X,0;\zeta)\exp\left(-\zeta X\right) = 1 - \frac{1}{3\zeta}[W(X) - W(-\infty)] + O\left(\zeta^{-2}\right),\tag{57}$$

i.e., W Xð Þ� Wð Þ¼ �∞ lim<sup>ζ</sup>!<sup>∞</sup> ½ � 3ζð Þ 1 � ψ exp ð Þ �ζX . Taking into account the functional dependence (56), we find the required one-soliton solution of the VPE in form

$$W(X,T) = 6\sqrt{3}\xi\_1 \frac{\partial}{\partial X} \ln\left[1 + \frac{\beta\_1}{2\sqrt{3}} \exp\left(\sqrt{3}\xi\_1 X - \frac{1}{\sqrt{3}\xi\_1}T\right)\right] + \text{const.}\tag{58}$$

Thus, for the example of the one-soliton solution, we have demonstrated the IST method.

#### 5.2 The direct spectral problem

Let us consider the principal aspects of the inverse scattering transform problem for a third-order equation. The inverse problem for certain third-order spectral equations has been considered by Kaup [33] and Caudrey [34, 35]. The time evolution of ψ is determined from (48) or (51).

Following the method described by Caudrey [34], the spectral equation (47) can be rewritten

$$\frac{\partial}{\partial X} \cdot \boldsymbol{\nu} = [\mathbf{A}(\boldsymbol{\zeta}) + \mathbf{B}(X, \boldsymbol{\zeta})] \cdot \boldsymbol{\nu} \tag{59}$$

with

$$\boldsymbol{\Psi} = \begin{pmatrix} \boldsymbol{\Psi} \\ \boldsymbol{\Psi}\_{\boldsymbol{X}} \\ \boldsymbol{\Psi}\_{\boldsymbol{X}\boldsymbol{X}} \end{pmatrix}, \qquad \mathbf{A} = \begin{pmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} \\ \boldsymbol{\lambda} & \mathbf{0} & \mathbf{0} \end{pmatrix}, \qquad \mathbf{B} = \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{W}\_{\boldsymbol{X}} & \mathbf{0} \end{pmatrix}. \tag{60}$$

The matrix A has eigenvalues λjð Þζ and left and right eigenvectors v~<sup>j</sup> ð Þζ and vjð Þζ , respectively. These quantities are defined through a spectral parameter λ as

$$\begin{aligned} \lambda\_j(\zeta) &= \alpha\_j \zeta, \quad \lambda\_j^3(\zeta) = \lambda, \\ \mathfrak{v}\_j(\zeta) &= \begin{pmatrix} \mathbf{1} \\ \lambda\_j(\zeta) \\ \lambda\_j^2(\zeta) \end{pmatrix}, \qquad \breve{\mathfrak{v}}\_j(\zeta) = \begin{pmatrix} \lambda\_j^2(\zeta) & \lambda\_j(\zeta) & \mathbf{1} \end{pmatrix}, \end{aligned} \tag{61}$$

#### Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

ξ1ð Þ¼ T ξ1ð Þ¼ 0 const:, <sup>β</sup>1ð Þ¼ <sup>T</sup> <sup>β</sup>1ð Þ <sup>0</sup> exp � <sup>1</sup>

that the spectrum λ in (47) remains constant if W Xð Þ ; T evolves according to Eq. (27). Therefore, as will be proved, the spectrum data evolve as in (70). In

3ζ

notations (77) and (78), from (70) we obtain the relations (56).

ln 1 <sup>þ</sup> <sup>β</sup><sup>1</sup>

2 ffiffiffi

asymptotic series in ζ�<sup>1</sup> to obtain

Research Advances in Chaos Theory

W Xð Þ¼ ; <sup>T</sup> <sup>6</sup> ffiffiffi

form

method.

be rewritten

with

ψ ¼

24

ψ ψX ψXX 1

CA, <sup>A</sup> <sup>¼</sup>

vjð Þ¼ ζ

0

B@

<sup>ψ</sup>ð Þ <sup>X</sup>; <sup>0</sup>; <sup>ζ</sup> exp ð Þ¼ �ζ<sup>X</sup> <sup>1</sup> � <sup>1</sup>

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup> ∂ ∂X

tion of ψ is determined from (48) or (51).

∂

0

B@

1 λjð Þζ λ2 <sup>j</sup> ð Þζ

<sup>λ</sup>jð Þ¼ <sup>ζ</sup> <sup>ω</sup>jζ, <sup>λ</sup><sup>3</sup>

0

BB@

5.2 The direct spectral problem

Below, the assumption of these relationships will be justified. Indeed, we know

The final step in IST method is to select the solution W Xð Þ ; T from (55) with ξ1ð Þ T , β1ð Þ T as in (56). According to Eq. (2.7) in [33], we expand ψð Þ X; T; ζ as an

i.e., W Xð Þ� Wð Þ¼ �∞ lim<sup>ζ</sup>!<sup>∞</sup> ½ � 3ζð Þ 1 � ψ exp ð Þ �ζX . Taking into account the functional dependence (56), we find the required one-soliton solution of the VPE in

<sup>3</sup> <sup>p</sup> exp ffiffiffi

Thus, for the example of the one-soliton solution, we have demonstrated the IST

Let us consider the principal aspects of the inverse scattering transform problem

Following the method described by Caudrey [34], the spectral equation (47) can

1

CA, <sup>B</sup> <sup>¼</sup>

ð Þ¼ <sup>ζ</sup> <sup>λ</sup><sup>2</sup>

for a third-order equation. The inverse problem for certain third-order spectral equations has been considered by Kaup [33] and Caudrey [34, 35]. The time evolu-

The matrix A has eigenvalues λjð Þζ and left and right eigenvectors v~<sup>j</sup>

<sup>j</sup> ð Þ¼ ζ λ,

CCA, <sup>v</sup>~<sup>j</sup>

1

vjð Þζ , respectively. These quantities are defined through a spectral parameter λ as

3 <sup>p</sup> <sup>ξ</sup>1<sup>X</sup> � <sup>1</sup>

� � � �

ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup> T � �

: (56)

þ const: (58)

<sup>½</sup>W Xð Þ� <sup>W</sup>ð Þ �<sup>∞</sup> � þ <sup>O</sup> <sup>ζ</sup>�<sup>2</sup> � �, (57)

ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup> T

<sup>∂</sup><sup>X</sup> <sup>ψ</sup> <sup>¼</sup> <sup>½</sup>Að Þþ <sup>ζ</sup> <sup>B</sup>ð Þ <sup>X</sup>; <sup>ζ</sup> � � <sup>ψ</sup> (59)

0

B@

<sup>j</sup> ð Þζ λjð Þζ 1 � �

000 000 0 �WX 0 1

CA: (60)

ð Þζ and

, (61)

where, as previously, <sup>ω</sup><sup>j</sup> <sup>¼</sup> <sup>e</sup>2πið Þ <sup>j</sup>�<sup>1</sup> <sup>=</sup><sup>3</sup> are the cube roots of 1 (<sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3). Obviously the λjð Þζ are distinct, and they and v~<sup>j</sup> ð Þζ and vjð Þζ are analytic throughout the complex ζ-plane.

The solution of the linear equation (47) (or equivalently (59)) has been obtained by Caudrey [34] in terms of Jost functions ϕ<sup>j</sup> ð Þ X; ζ which have the asymptotic behaviour:

$$\Phi\_{\dot{\jmath}}(X,\zeta) \coloneqq \exp\left\{-\lambda\_{\dot{\jmath}}(\zeta)X\right\} \phi\_{\dot{\jmath}}(X,\zeta) \to \upsilon\_{\dot{\jmath}}(\zeta) \quad \text{as} \ X \to -\infty. \tag{62}$$

Caudrey [34] showed how Eq. (59) can be solved by expressing it as a Fredholm integral equation.

The complex ζ-plane is to be divided into regions such that, in the interior of each region, the order of the numbers Reð Þ λið Þζ is fixed. As we pass from one region to another, this order changes, and hence, on a boundary between two regions, Reð Þ¼ <sup>λ</sup>ið Þ<sup>ζ</sup> Re <sup>λ</sup>jð Þ<sup>ζ</sup> � � for at least one pair <sup>i</sup> 6¼ <sup>j</sup>. The Jost function <sup>ϕ</sup><sup>j</sup> is regular throughout the complex ζ-plane apart from poles and finite singularities on the boundaries between the regions. At any point in the interior of any region of the complex ζ-plane, the solution of Eq. (59) is obtained by the relation (2.12) from [34]. It is the direct spectral problem.

#### 5.3 The spectral data

The information about the singularities of the Jost functions ϕjð Þ X; ζ reside in the spectral data. First let us consider the poles. It is assumed that a pole ζ ð Þk <sup>i</sup> in ϕið Þ X; ζ is simple, does not coincide with a pole of ϕ<sup>j</sup> ð Þ X; ζ and j 6¼ i and does not lie on a boundary between two regions. Then, as proven in [34], the residue is

$$\text{Res}\quad\phi\_i\left(\mathbf{X},\zeta\_i^{(k)}\right) = \sum\_{j=1}^n \nu\_{ij}^{(k)}\quad\phi\_j\left(\mathbf{X},\zeta\_i^{(k)}\right)\tag{63}$$

and it can be found because we know the solution (47) in any regular regions from solving the direct problem (see Section 5.2). Note that, for ϕ<sup>j</sup> X; ζ ð Þk i � �, the point ζ ð Þk <sup>i</sup> lies in the interior of a regular region. The quantities ζ ð Þk <sup>i</sup> and γ ð Þk ij constitute the discrete part of the spectral data.

Now we consider the singularities on the boundaries between regions. However, in order to simplify matters, we first make some observations. The solution of the spectral problem can be facilitated by using various symmetry properties. In view of (47), we need only consider the first elements of

$$\phi\_i(X,\zeta) = \begin{pmatrix} \phi\_i(X,\zeta) \\ \phi\_i(X,\zeta)\_X \\ \phi\_i(X,\zeta)\_{\text{XX}} \end{pmatrix},\tag{64}$$

whilst the symmetry

$$
\phi\_1(\mathbf{X}, \zeta/\alpha\_1) = \phi\_2(\mathbf{X}, \zeta/\alpha\_2) = \phi\_3(\mathbf{X}, \zeta/\alpha\_3) \tag{65}
$$

means we need only to consider ϕ1ð Þ X; ζ . In our case, for ϕ1ð Þ X; ζ , the complex ζ-plane is divided into four regions by two lines (see Figure 5) given by

<sup>Q</sup>1jð Þ¼ <sup>ζ</sup> <sup>1</sup>

Thus, the spectral data are

(48) at X ! ∞ together with (62)

the T-dependence is revealed as

ð Þk <sup>j</sup> ð Þ 0 ,

ð Þk

5.4 The inverse spectral problem

Φ1ð Þ¼ X; T; ζ 1 � ∑

27

þ 1 2πi ð ∑ 3 j¼2

B Xð Þ ; T; ζ and W Xð Þ ; T from the spectral data S.

K k¼1 ∑ 3 j¼2 γ ð Þk <sup>1</sup><sup>j</sup> ð Þ T

Figure 5 (for the lines (66), ξ sweeps from �∞ to þ∞).

the spectral data.

Loop-like Solitons

ζ ð Þk <sup>j</sup> ð Þ¼ T ζ

γ ð Þk <sup>1</sup><sup>j</sup> ð Þ¼ T γ

<sup>v</sup>~jð Þ� <sup>ζ</sup> vjð Þ<sup>ζ</sup> <sup>v</sup>~<sup>j</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86583

S ¼ ζ ð Þk <sup>1</sup> ; γ ð Þk

ð Þ� ζ

∞ð

exp <sup>λ</sup>1ð Þ� <sup>ζ</sup> <sup>λ</sup>jð Þ<sup>ζ</sup> � �<sup>z</sup> � �Bð Þ� <sup>z</sup>; <sup>ζ</sup> <sup>ϕ</sup>�

<sup>1</sup> ð Þ X; ζ dz:

(68)

(70)

�∞

The quantities Q1<sup>j</sup> ζ<sup>0</sup> ð Þ along all the boundaries constitute the continuum part of

One of the important features which is to be noted for the IST method is as follows. After the spectral data have been found from Bð Þ X; 0; ζ , i.e., at initial time, we need to seek the time evolution of the spectral data from Eq. (48). Analysing

> ð Þk 1 � � � � �<sup>1</sup>

The final step in the application of the IST method is to reconstruct B Xð Þ ; T; ζ from the evaluated spectral data. In the next section, we show how to do this.

The final procedure in IST method is that of the reconstruction of the matrix

exp λ<sup>j</sup> ζ

Eq. (71) contains the spectral data, namely, K poles with the quantities γ

the bound state spectrum as well as the functions Q1<sup>j</sup> ζ<sup>0</sup> ð Þ given along all the boundaries of regular regions for the continuous spectrum. The integral in (71) is along all the boundaries (see the dashed lines in Figure 5). The direction of integration is taken so that the side chosen to be Re <sup>λ</sup>1ð Þ� <sup>ζ</sup> <sup>λ</sup>jð Þ<sup>ζ</sup> � �<0 is shown by the arrows in

The spectral data define Φ1ð Þ X; ζ uniquely in the form (see Eq. (6.20) in [34]))

λ<sup>1</sup> ζ ð Þk 1 � � � <sup>λ</sup>1ð Þ<sup>ζ</sup>

<sup>Q</sup>1<sup>j</sup> <sup>T</sup>; <sup>ζ</sup><sup>0</sup> ð Þ exp <sup>λ</sup><sup>j</sup> <sup>ζ</sup><sup>0</sup> ð Þ� <sup>λ</sup><sup>1</sup> <sup>ζ</sup><sup>0</sup> ð Þ � �<sup>X</sup> � �

ð Þk 1 � � � <sup>λ</sup><sup>1</sup> <sup>ζ</sup>

ζ<sup>0</sup> � ζ

h i � � <sup>X</sup> n o

<sup>ϕ</sup>ið Þ¼ <sup>X</sup>; <sup>T</sup>; <sup>ζ</sup> exp �ð Þ <sup>3</sup>λið Þ<sup>ζ</sup> �<sup>1</sup>

<sup>Q</sup>1<sup>j</sup> <sup>T</sup>; <sup>ζ</sup><sup>0</sup> ð Þ¼ <sup>Q</sup>1<sup>j</sup> <sup>0</sup>; <sup>ζ</sup><sup>0</sup> ð Þ exp � <sup>3</sup>λ<sup>j</sup> <sup>ζ</sup><sup>0</sup> ð Þ � ��<sup>1</sup> <sup>þ</sup> <sup>3</sup>λ<sup>1</sup> <sup>ζ</sup><sup>0</sup> ð Þ ð Þ �<sup>1</sup> h i<sup>T</sup>

<sup>1</sup><sup>j</sup> ð Þ 0 exp � 3λ<sup>j</sup> ζ

<sup>1</sup><sup>j</sup> ; Q1<sup>j</sup> ζ<sup>0</sup> ð Þ; j ¼ 2; 3; k ¼ 1; 2; …; m

n o: (69)

T h iϕið Þ <sup>X</sup>; <sup>0</sup>; <sup>ζ</sup> ,

þ 3λ<sup>1</sup> ζ

� � � � �<sup>1</sup> � �<sup>T</sup> � �,

n o:

ð Þk 1

ð Þk 1

Φ�

Φ<sup>1</sup> X; T;ωjζ

<sup>1</sup> <sup>X</sup>; <sup>T</sup>;ωjζ<sup>0</sup> � �dζ<sup>0</sup>

ð Þk 1 � �

:

(71)

ð Þk <sup>1</sup><sup>j</sup> for

#### Figure 5.

The regular regions for Jost functions ϕ1(X, ζ) in the complex ζ-plane. The dashed lines determine the boundaries between regular regions. These lines are lines where the singularity functions Q1j(ζ<sup>0</sup> ) are given. The dotted lines are the lines where the poles appear.

$$\begin{array}{ll} \text{(i)} \quad \zeta' = o\_2 \mathfrak{z}, \quad \text{where} \quad \mathrm{Re}(\lambda\_1(\zeta)) = \mathrm{Re}(\lambda\_2(\zeta)),\\\\ \text{(ii)} \quad \zeta' = -o\_3 \mathfrak{z}, \quad \text{where} \quad \mathrm{Re}(\lambda\_1(\zeta)) = \mathrm{Re}(\lambda\_3(\zeta)), \end{array} \tag{66}$$

where ξ is real (see Figure 5). The singularity of ϕ1ð Þ X; ζ can appear only on these boundaries between the regular regions on the ζ-plane, and it is characterized by functions Q1<sup>j</sup> ζ<sup>0</sup> ð Þ at each fixed j 6¼ 1. We denote the limit of a quantity, as the boundary is approached, by the superfix � in according to the sign of Re <sup>λ</sup>1ð Þ� <sup>ζ</sup> <sup>λ</sup>jð Þ<sup>ζ</sup> (see Figure 5).

In [34] (see Eq. (3.14) there) the jump of ϕ1ð Þ X; ζ on the boundaries is calculated as

$$
\phi\_1^+(X,\zeta) - \phi\_1^-(X,\zeta) = \sum\_{j=2}^3 Q\_{\mathfrak{H}\_j^\circ}(\zeta)\phi\_j^-(X,\zeta),\tag{67}
$$

where, from (66), the sum is over the lines ζ<sup>0</sup> ¼ ω2ξ and ζ<sup>0</sup> ¼ �ω3ξ given by

$$\begin{array}{cccc} \text{(i)} & \zeta' = a\_2 \mathfrak{z}, & \text{with} & \mathcal{Q}\_{12}^{(1)}(\zeta') \neq \mathbf{0}, & \mathcal{Q}\_{13}^{(1)}(\zeta') \equiv \mathbf{0}, \\\\ \text{(ii)} & \zeta' = -a\_3 \mathfrak{z}, & \text{with} & \mathcal{Q}\_{12}^{(2)}(\zeta') \equiv \mathbf{0}, & \mathcal{Q}\_{13}^{(2)}(\zeta') \neq \mathbf{0}. \end{array}$$

The singularity functions Q1<sup>j</sup> ζ<sup>0</sup> ð Þ are determined by W Xð Þ ; 0 through the matrix Bð Þ X; ζ (60) (see Eq. (3.13) in [34])

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

$$Q\_{ij}(\zeta) = \frac{1}{\ddot{\boldsymbol{v}}\_j(\zeta) \cdot \boldsymbol{v}\_j(\zeta)} \cdot \ddot{\boldsymbol{v}}\_j(\zeta) \cdot \int\_{-\infty}^{\infty} \exp\left[ (\lambda\_1(\zeta) - \lambda\_j(\zeta))z \right] \mathbf{B}(z, \zeta) \cdot \phi\_1^-(\mathbf{X}, \zeta) \, dz. \tag{68}$$

The quantities Q1<sup>j</sup> ζ<sup>0</sup> ð Þ along all the boundaries constitute the continuum part of the spectral data.

Thus, the spectral data are

$$\mathcal{S} = \left\{ \zeta\_1^{(k)}, \gamma\_{1\dot{\jmath}}^{(k)}, \mathcal{Q}\_{\mathbb{A}\dot{\jmath}}(\zeta'); j = 2, 3, k = 1, 2, \ldots, m \right\}. \tag{69}$$

One of the important features which is to be noted for the IST method is as follows. After the spectral data have been found from Bð Þ X; 0; ζ , i.e., at initial time, we need to seek the time evolution of the spectral data from Eq. (48). Analysing (48) at X ! ∞ together with (62)

$$\phi\_i(\mathbf{X}, T, \zeta) = \exp\left[-(\mathbf{3}\lambda\_i(\zeta))^{-1}T\right] \phi\_i(\mathbf{X}, \mathbf{0}, \zeta),$$

the T-dependence is revealed as

$$\begin{aligned} \zeta\_j^{(k)}(T) &= \zeta\_j^{(k)}(\mathbf{0}),\\ \gamma\_{\mathbf{i}\mathbf{j}}^{(k)}(T) &= \gamma\_{\mathbf{i}\mathbf{j}}^{(k)}(\mathbf{0}) \exp\left\{ \left[ -\left( 3\dot{\lambda}\_j \Big(\zeta\_1^{(k)} \Big) \right)^{-1} + \left( 3\dot{\lambda}\_1 \Big(\zeta\_1^{(k)} \Big) \right)^{-1} \right] T \right\}, & \end{aligned} \tag{70}$$

$$\mathcal{Q}\_{\mathbf{i}\mathbf{j}}(T;\zeta') = \mathcal{Q}\_{\mathbf{i}\mathbf{j}}(\mathbf{0};\zeta') \exp\left\{ \left[ -\left( 3\dot{\lambda}\_{\mathbf{i}}(\zeta') \right)^{-1} + \left( 3\dot{\lambda}\_1 (\zeta') \right)^{-1} \right] T \right\}.$$

The final step in the application of the IST method is to reconstruct B Xð Þ ; T; ζ from the evaluated spectral data. In the next section, we show how to do this.

#### 5.4 The inverse spectral problem

The final procedure in IST method is that of the reconstruction of the matrix B Xð Þ ; T; ζ and W Xð Þ ; T from the spectral data S.

The spectral data define Φ1ð Þ X; ζ uniquely in the form (see Eq. (6.20) in [34]))

$$\begin{split} \boldsymbol{\Phi}\_{1}(\boldsymbol{X},T;\boldsymbol{\zeta}) &= \mathbf{1} - \sum\_{k=1}^{K} \sum\_{j=2}^{3} \boldsymbol{\gamma}\_{ij}^{(k)}(T) \frac{\exp\left\{ \left[ \boldsymbol{\lambda}\_{j} \left( \boldsymbol{\zeta}\_{1}^{(k)} \right) - \boldsymbol{\lambda}\_{1} \left( \boldsymbol{\zeta}\_{1}^{(k)} \right) \right] \boldsymbol{X} \right\}}{\boldsymbol{\lambda}\_{1} \left( \boldsymbol{\zeta}\_{1}^{(k)} \right) - \boldsymbol{\lambda}\_{1}(\boldsymbol{\zeta})} \boldsymbol{\Phi}\_{1} \left( \boldsymbol{X},T;\boldsymbol{\alpha}\_{\boldsymbol{\zeta}} \boldsymbol{\zeta}\_{1}^{(k)} \right) \\ &+ \frac{1}{2\pi \mathbf{i}} \int\_{j=2}^{3} \boldsymbol{\mathcal{Q}}\_{\text{ij}}(T;\boldsymbol{\zeta}') \frac{\exp\left\{ \left[ \boldsymbol{\lambda}\_{j} (\boldsymbol{\zeta}') - \boldsymbol{\lambda}\_{1} (\boldsymbol{\zeta}') \right] \boldsymbol{X} \right\}}{\boldsymbol{\zeta}' - \boldsymbol{\zeta}} \boldsymbol{\Phi}\_{1}^{-}(\boldsymbol{X},T;\boldsymbol{\alpha}\_{\boldsymbol{\zeta}} \boldsymbol{\zeta}') d\boldsymbol{\zeta}'. \end{split} \tag{71}$$

Eq. (71) contains the spectral data, namely, K poles with the quantities γ ð Þk <sup>1</sup><sup>j</sup> for the bound state spectrum as well as the functions Q1<sup>j</sup> ζ<sup>0</sup> ð Þ given along all the boundaries of regular regions for the continuous spectrum. The integral in (71) is along all the boundaries (see the dashed lines in Figure 5). The direction of integration is taken so that the side chosen to be Re <sup>λ</sup>1ð Þ� <sup>ζ</sup> <sup>λ</sup>jð Þ<sup>ζ</sup> � �<0 is shown by the arrows in Figure 5 (for the lines (66), ξ sweeps from �∞ to þ∞).

ð Þi ζ<sup>0</sup> ¼ ω2ξ, where Reð Þ¼ λ1ð Þζ Reð Þ λ2ð Þζ ,

The regular regions for Jost functions ϕ1(X, ζ) in the complex ζ-plane. The dashed lines determine the boundaries between regular regions. These lines are lines where the singularity functions Q1j(ζ<sup>0</sup>

ð Þii ζ<sup>0</sup> ¼ �ω3ξ, where Reð Þ¼ λ1ð Þζ Reð Þ λ3ð Þζ ,

where ξ is real (see Figure 5). The singularity of ϕ1ð Þ X; ζ can appear only on these boundaries between the regular regions on the ζ-plane, and it is characterized

In [34] (see Eq. (3.14) there) the jump of ϕ1ð Þ X; ζ on the boundaries is calculated as

3 j¼2

Q1<sup>j</sup>ð Þζ ϕ�

<sup>12</sup> <sup>ζ</sup><sup>0</sup> ð Þ 6¼ <sup>0</sup>, Qð Þ<sup>1</sup>

<sup>12</sup> <sup>ζ</sup><sup>0</sup> ð Þ� <sup>0</sup>, Qð Þ<sup>2</sup>

<sup>j</sup> ð Þ X; ζ , (67)

<sup>13</sup> ζ<sup>0</sup> ð Þ� 0,

<sup>13</sup> ζ<sup>0</sup> ð Þ 6¼ 0:

<sup>1</sup> ð Þ¼ X; ζ ∑

where, from (66), the sum is over the lines ζ<sup>0</sup> ¼ ω2ξ and ζ<sup>0</sup> ¼ �ω3ξ given by

The singularity functions Q1<sup>j</sup> ζ<sup>0</sup> ð Þ are determined by W Xð Þ ; 0 through the matrix

by functions Q1<sup>j</sup> ζ<sup>0</sup> ð Þ at each fixed j 6¼ 1. We denote the limit of a quantity, as the boundary is approached, by the superfix � in according to the sign of

Re <sup>λ</sup>1ð Þ� <sup>ζ</sup> <sup>λ</sup>jð Þ<sup>ζ</sup> (see Figure 5).

The dotted lines are the lines where the poles appear.

Research Advances in Chaos Theory

Figure 5.

Bð Þ X; ζ (60) (see Eq. (3.13) in [34])

26

ϕ<sup>þ</sup>

<sup>1</sup> ð Þ� X; ζ ϕ�

ð Þ<sup>i</sup> <sup>ζ</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup>2ξ, with <sup>Q</sup>ð Þ<sup>1</sup>

ð Þii <sup>ζ</sup><sup>0</sup> ¼ �ω3ξ, with <sup>Q</sup>ð Þ<sup>2</sup>

(66)

) are given.

It is necessary to note that we should carry out the integration along the lines ω2ð Þ ξ þ iε and �ω3ð Þ ξ þ iε with ε>0. In this case condition (62) is satisfied. Passing to the limit ε ! 0, we can obtain the solution which does not satisfy condition (62). However, for any finite ε>0, the restricted region on X can be determined where the solution associated with a finite ε> 0 (for which the condition (62) is valid) and the solution associated with ε ¼ 0 are sufficiently close to each other. In this sense, taking the integration at ε ¼ 0, we remain within the inverse scattering theory [34], and so condition (62) can be omitted. The solution obtained at ε ¼ 0 can be extended to sufficiently large finite X. Thus, we will interpret the solution obtained at ε ¼ 0 as the solution of the VPE (27) which is valid for arbitrary but finite X.

As follows from Eqs. (2.12), (2.13), (2.36) and (2.37) of [33], ψXð Þζ is related to

ð Þ1

<sup>1</sup> . Hence, the poles appear in pairs, ζ

<sup>1</sup> ¼ �ω2, where n is the pair number.

sum is taken in (76). For the pair n nð Þ ¼ 1; 2; …; N we have the properties

<sup>1</sup> ¼ iω2ξn, ð Þii ζ

with the condition (2.33) from [33]. These relationships are also similar to

ð Þk <sup>12</sup> and γ

ϕ1ð Þ X; �ω2ζ and ϕ1,Xð Þ X; �ω2ζ also have poles here, whilst the functions

form (73) into Eq. (76) and letting X ! �∞, we have the ratio γ

<sup>13</sup> ¼ 0. Therefore, the properties of γ

ð Þ 2n�1

ð Þ 2n

ð Þi γ

ð Þii γ

3 j¼2 γ ð Þk

Φ1ð Þ¼ X; T; ζ 1 � ∑

2N k¼1

Ψkð Þ¼ X; T ∑

ð Þk

Boussinesq equation (see Eqs. (6.24) and (6.25) in [34]). Indeed, by considering

ϕ1ð Þ X; �ω3ζ and ϕ1,Xð Þ X; �ω3ζ do not have poles here. Substituting ϕ1ð Þ X; ζ in the

<sup>12</sup> ¼ ω2βk, γ

<sup>12</sup> ¼ 0, γ

where, as it will be proved below, β<sup>k</sup> is real when U ¼ WX is real.

<sup>1</sup><sup>j</sup> ð Þ T exp λ<sup>j</sup> ζ

2N k¼1

From (72) and (80), it may be shown that (cf. Eq. (6.38) in [34])

exp �λ<sup>1</sup> ζ

ð Þk

ð Þ 2n�1

Let us consider N pairs of poles, i.e., in all there are K ¼ 2N poles over which the

Since U is real and λ is imaginary, ξ<sup>k</sup> is real. The relationships (77) are in line

<sup>X</sup>ð Þ �ζ . In the usual manner, using the adjoint states and Eq. (14)

<sup>1</sup> is a pole of ϕ1ð Þ X; ζ , then there is a pole either at

ð Þ 2n

<sup>1</sup><sup>j</sup> turns out to be different from ~γ

ð Þk

ð Þ 2n�1 <sup>13</sup> ¼ 0,

n oΦ<sup>1</sup> <sup>X</sup>; <sup>T</sup>;ωj<sup>ζ</sup>

ð Þk 1 � �<sup>X</sup> n o

ð Þ 2n

ð Þk 1 � �<sup>X</sup>

we may rewrite the relationship (73) as (see, for instance, Eqs. (6.33) and (6.34)

exp �λ<sup>1</sup> ζ

n oΨkð Þ¼ <sup>X</sup>; <sup>T</sup> <sup>3</sup> <sup>∂</sup>

λ<sup>1</sup> ζ ð Þk 1 � � � <sup>λ</sup>1ð Þ<sup>ζ</sup>

ð Þk 1 � �<sup>X</sup>

<sup>1</sup> of the pair n and using the relation (73),

<sup>13</sup> . In this case the functions ϕ1,Xð Þ X; ζ ,

ð Þ 2n <sup>13</sup> =γ

ij should be defined by the

<sup>13</sup> <sup>¼</sup> <sup>ω</sup>3βk, (78)

ð Þk 1 � �, (79)

Ψkð Þ X; T : (80)

<sup>∂</sup><sup>X</sup> ln det ð Þ M Xð Þ ; <sup>T</sup> : (81)

ð Þ1

ð Þ1

ð Þ 2n�1 <sup>1</sup> and ζ

<sup>1</sup> , since �ω3ζ

<sup>1</sup> ¼ �iω3ξn: (77)

ð Þ 2n

<sup>1</sup> . Then, as follows from (76), �ω3ζ

<sup>1</sup> (if ϕ1ð Þ X; �ω3ζ has a

ð Þ2 1

ð Þ2 1 ¼

<sup>1</sup> , under the

ð Þk <sup>1</sup><sup>j</sup> for the

ð Þ 2n�1 <sup>12</sup> ¼ ω<sup>2</sup>

<sup>3</sup> <sup>p</sup> ½ � <sup>ϕ</sup>1Xð Þ <sup>X</sup>; �ω2<sup>ζ</sup> <sup>ϕ</sup>1ð Þ� <sup>X</sup>; �ω3<sup>ζ</sup> <sup>ϕ</sup>1Xð Þ <sup>X</sup>; �ω3<sup>ζ</sup> <sup>ϕ</sup>1ð Þ <sup>X</sup>; �ω2<sup>ζ</sup> : (76)

ð Þ2 <sup>1</sup> ¼ �ω3ζ

the adjoint states ψ<sup>A</sup>

Loop-like Solitons

<sup>ϕ</sup>1Xð Þ¼ <sup>X</sup>; <sup>ζ</sup> <sup>i</sup>

ζ ð Þ2 <sup>1</sup> ¼ �ω2ζ

�ω3ð Þ �ω<sup>2</sup> ζ

condition ζ

and γ ð Þ 2n <sup>12</sup> ¼ γ

relationships

By defining

in [34])

29

ffiffiffi

DOI: http://dx.doi.org/10.5772/intechopen.86583

ð Þ 2n�1

Eqs. (6.24) and (6.25) in [34], whilst γ

(76) in the vicinity of the first pole ζ

one can obtain a relation between γ

ð Þ 2n�1

W Xð Þ� ; T Wð Þ¼� �∞ 3 ∑

ð Þi ζ

It is easily seen that if ζ

ð Þ1

ð Þ1 <sup>1</sup> ¼ ζ ð Þ1

ð Þ 2n <sup>1</sup> =ζ

pole). For definiteness let ζ

from [35] and Eq. (2.37) from [33], one can obtain

ð Þ1

<sup>1</sup> (if ϕ1ð Þ X; �ω2ζ has a pole) or at ζ

ð Þ2 <sup>1</sup> ¼ �ω2ζ

should be a pole. However, this pole coincides with pole ζ

ð Þ 2n�1

By choosing appropriate values for ζ, the left-hand side in (71) can be Φ<sup>1</sup> X; T;ωjζ ð Þk 1 � �, or by allowing <sup>ζ</sup> to approach the boundaries from the appropriate sides, the left-hand side can be Φ� <sup>1</sup> <sup>X</sup>; <sup>T</sup>;ωjζ<sup>0</sup> � �. We acquire a set of linear matrix/ Fredholm equations in the unknowns Φ<sup>1</sup> X; T;ωjζ ð Þk 1 � � and <sup>Φ</sup>� <sup>1</sup> <sup>X</sup>; <sup>T</sup>;ωjζ<sup>0</sup> � �. The solution of this equation system enables one to define Φ1ð Þ X; T; ζ from (71).

By knowing Φ1ð Þ X; T; ζ , we can take extra information into account, namely, that the expansion of <sup>Φ</sup>1ð Þ <sup>X</sup>; <sup>T</sup>; <sup>ζ</sup> as an asymptotic series in <sup>λ</sup>�<sup>1</sup> <sup>1</sup> ð Þζ connects with W Xð Þ ; T as follows (cf. Eq. (2.7) in [33]):

$$\Phi\_1(X, T; \zeta) = 1 - \frac{1}{3\lambda\_1(\zeta)} [\mathcal{W}(X, T) - \mathcal{W}(-\infty)] + O\left(\lambda\_1^{-2}(\zeta)\right). \tag{72}$$

Consequently, the solution W Xð Þ ; T and the matrix B Xð Þ ; T; ζ can be reconstructed from the spectral data.

#### 6. The interaction of the loop-like solitons

We will discuss the exact N-soliton solution of the VPE via the inverse scattering method [24]. To do this we consider (71) with Q1<sup>j</sup>ð Þ� ζ 0. Then there is only the bound state spectrum which is associated with the soliton solutions.

Let the bound state spectrum be defined by K poles. The relation (71) is reduced to the form

$$\Phi\_1(X, T; \zeta) = \mathbf{1} - \sum\_{k=1}^K \sum\_{j=2}^3 \gamma\_{\mathbf{i}\_j}^{(k)}(T) \frac{\exp\left\{ \left[ \lambda\_{\mathbf{i}} \left( \zeta\_1^{(k)} \right) - \lambda\_{\mathbf{i}} \left( \zeta\_1^{(k)} \right) \right] X \right\}}{\lambda\_1 \left( \zeta\_1^{(k)} \right) - \lambda\_{\mathbf{i}}(\zeta)} \Phi\_1 \left( X, T; \alpha\_{\mathbf{j}} \zeta\_1^{(k)} \right). \tag{73}$$

Eq. (73) involves the spectral data, namely, the poles ζ ð Þk <sup>1</sup> and the quantities γ ð Þk 1j . First we will prove that Reλ ¼ 0 for compact support. From Eq. (47) we have

$$(\psi\_X)\_{\text{XXX}} + (U\psi\_X)\_X - \lambda \psi\_X = \mathbf{0},\tag{74}$$

and together with Eq. (47), this enables us to write

$$\frac{\partial}{\partial X} \left( \frac{\partial^2}{\partial X^2} \boldsymbol{\nu}\_X \boldsymbol{\nu}^\* - 3 \boldsymbol{\nu}\_{XX} \boldsymbol{\nu}\_X^\* + U \boldsymbol{\nu}\_X \boldsymbol{\nu}^\* \right) - 2 \text{Re} \lambda \boldsymbol{\nu}\_X \boldsymbol{\nu}^\* = \mathbf{0}. \tag{75}$$

Integrating Eq. (75) over all values of X, we obtain that, for compact support, <sup>R</sup>e<sup>λ</sup> <sup>¼</sup> 0 since, in the general case, <sup>Ð</sup> <sup>∞</sup> �<sup>∞</sup> <sup>ψ</sup>X<sup>ψ</sup> <sup>∗</sup> dX 6¼ 0.

It is necessary to note that we should carry out the integration along the lines ω2ð Þ ξ þ iε and �ω3ð Þ ξ þ iε with ε>0. In this case condition (62) is satisfied. Passing to the limit ε ! 0, we can obtain the solution which does not satisfy condition (62). However, for any finite ε>0, the restricted region on X can be determined where the solution associated with a finite ε> 0 (for which the condition (62) is valid) and the solution associated with ε ¼ 0 are sufficiently close to each other. In this sense, taking the integration at ε ¼ 0, we remain within the inverse scattering theory [34],

, or by allowing ζ to approach the boundaries from the appropriate

ð Þk 1 � �

<sup>1</sup> <sup>X</sup>; <sup>T</sup>;ωjζ<sup>0</sup> � �. We acquire a set of linear matrix/

and Φ�

<sup>1</sup> <sup>X</sup>; <sup>T</sup>;ωjζ<sup>0</sup> � �. The

<sup>1</sup> ð Þζ connects with

<sup>1</sup> ð Þ<sup>ζ</sup> � �: (72)

Φ<sup>1</sup> X; T;ωjζ

<sup>1</sup> and the quantities γ

� 2ReλψX<sup>ψ</sup> <sup>∗</sup> <sup>¼</sup> <sup>0</sup>: (75)

ð Þk 1 � �

:

(73)

ð Þk 1j .

and so condition (62) can be omitted. The solution obtained at ε ¼ 0 can be extended to sufficiently large finite X. Thus, we will interpret the solution obtained at ε ¼ 0 as the solution of the VPE (27) which is valid for arbitrary but finite X. By choosing appropriate values for ζ, the left-hand side in (71) can be

solution of this equation system enables one to define Φ1ð Þ X; T; ζ from (71).

Consequently, the solution W Xð Þ ; T and the matrix B Xð Þ ; T; ζ can be

bound state spectrum which is associated with the soliton solutions.

Eq. (73) involves the spectral data, namely, the poles ζ

and together with Eq. (47), this enables us to write

� �

<sup>∂</sup>X<sup>2</sup> <sup>ψ</sup>X<sup>ψ</sup> <sup>∗</sup> � <sup>3</sup>ψXX<sup>ψ</sup> <sup>∗</sup>

By knowing Φ1ð Þ X; T; ζ , we can take extra information into account, namely,

<sup>3</sup>λ1ð Þ<sup>ζ</sup> <sup>½</sup>W Xð Þ� ; <sup>T</sup> <sup>W</sup>ð Þ �<sup>∞</sup> � þ <sup>O</sup> <sup>λ</sup>�<sup>2</sup>

We will discuss the exact N-soliton solution of the VPE via the inverse scattering method [24]. To do this we consider (71) with Q1<sup>j</sup>ð Þ� ζ 0. Then there is only the

Let the bound state spectrum be defined by K poles. The relation (71) is reduced

λ<sup>1</sup> ζ ð Þk 1 � �

<sup>X</sup> <sup>þ</sup> <sup>U</sup>ψX<sup>ψ</sup> <sup>∗</sup>

Integrating Eq. (75) over all values of X, we obtain that, for compact support,

�<sup>∞</sup> <sup>ψ</sup>X<sup>ψ</sup> <sup>∗</sup> dX 6¼ 0.

ð Þk 1 � �

� λ<sup>1</sup> ζ ð Þk 1

� λ1ð Þζ

ð Þ ψ<sup>X</sup> XXX þ ð Þ Uψ<sup>X</sup> <sup>X</sup> � λψ<sup>X</sup> ¼ 0, (74)

X

ð Þk

h i � �

n o

exp λ<sup>j</sup> ζ

First we will prove that Reλ ¼ 0 for compact support. From Eq. (47) we have

Φ<sup>1</sup> X; T;ωjζ

to the form

28

Φ1ð Þ¼ X; T; ζ 1 � ∑

∂ ∂X ∂2

<sup>R</sup>e<sup>λ</sup> <sup>¼</sup> 0 since, in the general case, <sup>Ð</sup> <sup>∞</sup>

ð Þk 1 � �

Research Advances in Chaos Theory

sides, the left-hand side can be Φ�

W Xð Þ ; T as follows (cf. Eq. (2.7) in [33]):

<sup>Φ</sup>1ð Þ¼ <sup>X</sup>; <sup>T</sup>; <sup>ζ</sup> <sup>1</sup> � <sup>1</sup>

reconstructed from the spectral data.

6. The interaction of the loop-like solitons

K k¼1 ∑ 3 j¼2 γ ð Þk <sup>1</sup><sup>j</sup> ð Þ T

Fredholm equations in the unknowns Φ<sup>1</sup> X; T;ωjζ

that the expansion of <sup>Φ</sup>1ð Þ <sup>X</sup>; <sup>T</sup>; <sup>ζ</sup> as an asymptotic series in <sup>λ</sup>�<sup>1</sup>

As follows from Eqs. (2.12), (2.13), (2.36) and (2.37) of [33], ψXð Þζ is related to the adjoint states ψ<sup>A</sup> <sup>X</sup>ð Þ �ζ . In the usual manner, using the adjoint states and Eq. (14) from [35] and Eq. (2.37) from [33], one can obtain

$$\phi\_{1X}(X,\zeta) = \frac{\mathbf{i}}{\sqrt{3}} \left[ \phi\_{1X}(X, -a\nu\_2\zeta)\phi\_1(X, -a\nu\_3\zeta) - \phi\_{1X}(X, -a\nu\_3\zeta)\phi\_1(X, -a\nu\_2\zeta) \right]. \tag{76}$$

It is easily seen that if ζ ð Þ1 <sup>1</sup> is a pole of ϕ1ð Þ X; ζ , then there is a pole either at ζ ð Þ2 <sup>1</sup> ¼ �ω2ζ ð Þ1 <sup>1</sup> (if ϕ1ð Þ X; �ω2ζ has a pole) or at ζ ð Þ2 <sup>1</sup> ¼ �ω3ζ ð Þ1 <sup>1</sup> (if ϕ1ð Þ X; �ω3ζ has a pole). For definiteness let ζ ð Þ2 <sup>1</sup> ¼ �ω2ζ ð Þ1 <sup>1</sup> . Then, as follows from (76), �ω3ζ ð Þ2 1 should be a pole. However, this pole coincides with pole ζ ð Þ1 <sup>1</sup> , since �ω3ζ ð Þ2 1 ¼ �ω3ð Þ �ω<sup>2</sup> ζ ð Þ1 <sup>1</sup> ¼ ζ ð Þ1 <sup>1</sup> . Hence, the poles appear in pairs, ζ ð Þ 2n�1 <sup>1</sup> and ζ ð Þ 2n <sup>1</sup> , under the condition ζ ð Þ 2n <sup>1</sup> =ζ ð Þ 2n�1 <sup>1</sup> ¼ �ω2, where n is the pair number.

Let us consider N pairs of poles, i.e., in all there are K ¼ 2N poles over which the sum is taken in (76). For the pair n nð Þ ¼ 1; 2; …; N we have the properties

$$\text{(i)}\quad \zeta\_1^{(2n-1)} = \text{i}\alpha\_2 \xi\_n. \qquad \text{(ii)}\quad \zeta\_1^{(2n)} = -\text{i}\alpha\_3 \xi\_n. \tag{77}$$

Since U is real and λ is imaginary, ξ<sup>k</sup> is real. The relationships (77) are in line with the condition (2.33) from [33]. These relationships are also similar to Eqs. (6.24) and (6.25) in [34], whilst γ ð Þk <sup>1</sup><sup>j</sup> turns out to be different from ~γ ð Þk <sup>1</sup><sup>j</sup> for the Boussinesq equation (see Eqs. (6.24) and (6.25) in [34]). Indeed, by considering (76) in the vicinity of the first pole ζ ð Þ 2n�1 <sup>1</sup> of the pair n and using the relation (73), one can obtain a relation between γ ð Þk <sup>12</sup> and γ ð Þk <sup>13</sup> . In this case the functions ϕ1,Xð Þ X; ζ , ϕ1ð Þ X; �ω2ζ and ϕ1,Xð Þ X; �ω2ζ also have poles here, whilst the functions ϕ1ð Þ X; �ω3ζ and ϕ1,Xð Þ X; �ω3ζ do not have poles here. Substituting ϕ1ð Þ X; ζ in the form (73) into Eq. (76) and letting X ! �∞, we have the ratio γ ð Þ 2n <sup>13</sup> =γ ð Þ 2n�1 <sup>12</sup> ¼ ω<sup>2</sup> and γ ð Þ 2n <sup>12</sup> ¼ γ ð Þ 2n�1 <sup>13</sup> ¼ 0. Therefore, the properties of γ ð Þk ij should be defined by the relationships

$$\begin{aligned} \text{(i)} \qquad & \begin{aligned} \gamma\_{12}^{(2n-1)} &= o\_2 \beta\_k, & \gamma\_{13}^{(2n-1)} &= \mathbf{0}, \\ \gamma\_{12}^{(2n)} &= \mathbf{0}, & \gamma\_{13}^{(2n)} &= o\_3 \beta\_k. \end{aligned} \tag{78}$$

where, as it will be proved below, β<sup>k</sup> is real when U ¼ WX is real. By defining

$$\Psi\_k(\mathbf{X}, T) = \sum\_{j=2}^{3} \mathbf{y}\_{\mathbf{j}}^{(k)}(T) \exp\left\{\mathbb{A}\_{\mathbf{j}}\Big(\boldsymbol{\zeta}\_1^{(k)}\Big) \mathbf{X}\right\} \Phi\_1\Big(\mathbf{X}, T; \boldsymbol{\alpha}\_{\mathbf{j}}\boldsymbol{\zeta}\_1^{(k)}\Big),\tag{79}$$

we may rewrite the relationship (73) as (see, for instance, Eqs. (6.33) and (6.34) in [34])

$$\Phi\_1(X, T; \zeta) = \mathbf{1} - \sum\_{k=1}^{2N} \frac{\exp\left\{-\lambda\_1 \binom{k}{1} X\right\}}{\lambda\_1 \binom{k}{1} - \lambda\_1(\zeta)} \Psi\_k(X, T). \tag{80}$$

From (72) and (80), it may be shown that (cf. Eq. (6.38) in [34])

$$\mathcal{W}(X,T) - \mathcal{W}(-\infty) = -3 \sum\_{k=1}^{2N} \exp\left\{-\lambda\_1 \binom{k}{1} X\right\} \Psi\_k(X,T) = 3 \frac{\partial}{\partial X} \ln\left(\det M(X,T)\right). \tag{81}$$

The 2N � 2N matrix M Xð Þ ; T is defined as in relationship (6.36) in [34] by

U Xð Þ¼ ; <sup>T</sup> <sup>9</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86583

where

Loop-like Solitons

qi ¼ exp

and <sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup>

the loop-like solitons.

with respect to x, we find that Ð <sup>∞</sup>

in (b).

31

ð Þ v<sup>1</sup> þ v<sup>2</sup> =2.

integrating with respect to x, we obtain Ð <sup>∞</sup>

ffiffiffi 3 p

<sup>2</sup> ln <sup>β</sup>i=<sup>2</sup> ffiffiffi

<sup>2</sup> <sup>ξ</sup><sup>i</sup> <sup>X</sup> � <sup>T</sup>

<sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>i</sup>

6.2 The two-loop-like solitons of the VE

3ξ<sup>2</sup> i

!

" #

2 ξ2 1sinh �<sup>2</sup>

η, η ¼

4 � 4 matrix. We will not give the explicit form here, but we find that

det M Xð Þ¼ ; <sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

þ α<sup>i</sup>

VPE as found by the IST method is given by (81) together with (87).

so that det M Xð Þ ; T is a perfect square for arbitrary N.

rc ¼ 0:88867 in that we have a different form of the phase shift:

smaller loop soliton is not shifted by the interaction.

a. For rc <r<1, δ<sup>1</sup> < 0 so the smaller loop soliton is shifted backwards.

c. For 0 <r<rc, δ<sup>1</sup> >0 so the smaller loop soliton is shifted forwards.

b.For r ¼ rc, where rc ¼ 0:88867 is the root of ln b þ 3=r ¼ 0, δ<sup>1</sup> ¼ 0, so the

At first sight it might seem that the behaviour in (b) and (c) contradicts conservation of 'momentum'. That this is not so is justified as follows. By integrating (9)

of each soliton is zero, and 'momentum' is conserved whatever δ<sup>1</sup> and δ<sup>2</sup> may be. In particular δ<sup>1</sup> and δ<sup>2</sup> may have the same sign as in (c), or one of them may be zero as

Cases (a), (b) and (c) are illustrated in Figures 6–8, respectively; in these figures u is plotted against x for various values of t. For convenience in the figures, the interactions of solitons are shown in coordinates moving with speed

�<sup>∞</sup> udx <sup>¼</sup> 0; also, by multiplying (9) by <sup>x</sup> and

�<sup>∞</sup> xudx <sup>¼</sup> 0. Thus, in <sup>x</sup>-<sup>t</sup> space, the 'mass'

Let us now consider the two-soliton solution of the VPE. In this case M Xð Þ ; T is a

<sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

, b<sup>2</sup> <sup>¼</sup> <sup>ξ</sup><sup>2</sup> � <sup>ξ</sup><sup>1</sup>

Finally we note that comparison of (81) with W ¼ 6 lnð Þf <sup>X</sup> from (42) shows that

We discuss the two-loop soliton solution of the VE in more detail. Let us consider what happens in x-t space. The relations (20), (25) and (29) determine the solutions in x-t throughout the solutions in X-T. In these coordinates x-t, we have

The shifts, δi, of the two-loop solitons u<sup>1</sup> and u<sup>2</sup> in the positive x-direction due to the interaction may be computed as follows. The larger loop soliton is always shifted forwards by the interaction. However, for smaller u<sup>2</sup> with r ¼ ξ1=ξ2, there is a value

� �<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> q2 1q2 2

ξ<sup>2</sup> þ ξ<sup>1</sup> � �<sup>2</sup> ξ<sup>2</sup>

� � are arbitrary constants. The two-soliton solution to the

ffiffiffi 3 p

<sup>2</sup> <sup>ξ</sup><sup>1</sup> <sup>X</sup> � <sup>T</sup>

3ξ<sup>2</sup> 1

<sup>1</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup>

ξ2 <sup>1</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup>

ln det ð Þ¼ M Xð Þ ; T 2ln ð Þf : (89)

<sup>2</sup> � ξ1ξ<sup>2</sup>

<sup>2</sup> þ ξ1ξ<sup>2</sup>

þ α1: (86)

, (87)

, (88)

� �

$$M\_{kl}(X,T) = \delta\_{kl} - \sum\_{j=2}^{3} \mathbf{r}\_{lj}^{(k)}(0) \frac{\exp\left\{ \left[ -\left( 3\boldsymbol{\lambda}\_{l} \left( \boldsymbol{\xi}\_{1}^{(k)} \right) \right)^{-1} + \left( 3\boldsymbol{\lambda}\_{l} \left( \boldsymbol{\xi}\_{1}^{(k)} \right) \right)^{-1} \right] T + \left( \boldsymbol{\lambda}\_{l} \left( \boldsymbol{\xi}\_{1}^{(k)} \right) - \boldsymbol{\lambda}\_{l} \left( \boldsymbol{\xi}\_{1}^{(l)} \right) \right) X \right\}}{\boldsymbol{\lambda}\_{l}^{(k)}(\boldsymbol{\xi}\_{1}^{(k)}) - \boldsymbol{\lambda}\_{l} \left( \boldsymbol{\xi}\_{1}^{(l)} \right)} \tag{82}$$

and

$$\begin{aligned} n &= 1, & 2, \dots, N, \\ \lambda\_1 \left( \boldsymbol{\zeta}\_1^{(2n-1)} \right) &= i\alpha\_2 \boldsymbol{\xi}\_n, & \lambda\_2 \left( \boldsymbol{\zeta}\_1^{(2n-1)} \right) &= i\alpha\_3 \boldsymbol{\xi}\_n, & \boldsymbol{\gamma}\_{12}^{(2n-1)} &= \alpha\_2 \boldsymbol{\beta}\_n, & \boldsymbol{\gamma}\_{13}^{(2n-1)} &= \mathbf{0}, \\ \lambda\_1 \left( \boldsymbol{\zeta}\_1^{(2n)} \right) &= -i\alpha\_3 \boldsymbol{\xi}\_n, & \lambda\_3 \left( \boldsymbol{\zeta}\_1^{(2n)} \right) &= -i\alpha\_2 \boldsymbol{\xi}\_n, & \boldsymbol{\gamma}\_{12}^{(2n)} &= \mathbf{0}, & \boldsymbol{\gamma}\_{13}^{(2n)} &= \alpha\_3 \boldsymbol{\beta}\_n. \end{aligned}$$

For the N-soliton solution, there are N arbitrary constants ξ<sup>n</sup> and N arbitrary constants βn.

The final result for the N-soliton solution of the VPE is defined by relationship (81) with (82).

#### 6.1 Examples of one- and two-soliton solutions of the VPE

In order to obtain the one-soliton solution of the VPE (27)

$$\boldsymbol{W}\_{\boldsymbol{X}\boldsymbol{X}\boldsymbol{T}} + (\mathbf{1} + \boldsymbol{W}\_{\boldsymbol{T}}) \boldsymbol{W}\_{\boldsymbol{X}} = \mathbf{0},$$

we need first to calculate the 2 � 2 matrix M Xð Þ ; T according to (82) with N ¼ 1. We find that the matrix is

$$\begin{pmatrix} 1 - \frac{\alpha \rho\_1^{\varepsilon} \mathbb{I}\_1}{\sqrt{3} \xi\_1} \exp\left[ \sqrt{3} \xi\_1 X - \left( \sqrt{3} \xi\_1 \right)^{-1} T \right] & \frac{\mathrm{i} \alpha \beta\_1 \mathbb{I}\_1}{2 \xi\_1} \exp\left[ 2 \mathrm{i} \alpha \varphi\_1 X - \left( \sqrt{3} \xi\_1 \right)^{-1} T \right] \\\\ \frac{-\mathrm{i} \alpha \beta\_1}{2 \xi\_1} \exp\left[ -2 \mathrm{i} \alpha \varphi\_1 X - \left( \sqrt{3} \xi\_1 \right)^{-1} T \right] & 1 - \frac{\alpha \beta\_1 \mathbb{I}\_1}{\sqrt{3} \xi\_1} \exp\left[ \sqrt{3} \xi\_1 X - \left( \sqrt{3} \xi\_1 \right)^{-1} T \right] \end{pmatrix} \tag{83}$$

and its determinant is

$$\det M(X, T) = \left\{ \mathbf{1} + \frac{\beta\_1}{2\sqrt{3}\xi\_1} \exp\left[ \sqrt{3}\xi\_1 \left( X - \frac{T}{3\xi\_1^2} \right) \right] \right\}^2. \tag{84}$$

Consequently, from Eq. (81) we have the one-soliton solution of the VPE

$$U(X,T) = W\_X(X,T) = \frac{9}{2} \xi\_1^2 \text{sech}^2\left[\frac{\sqrt{3}}{2}\xi\_1\left(X - \frac{T}{3\xi\_1^2}\right) + a\_1\right],\tag{85}$$

where <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ln <sup>β</sup>1=<sup>2</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup> � � is an arbitrary constant. Since U is real, it follows from (85) that β<sup>1</sup> is real. Note that with β1=ξ<sup>1</sup> <0 we have the real solution in the form of the singular soliton [41].

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

$$U(X,T) = \frac{9}{2} \xi\_1^2 \sinh^{-2} \eta, \quad \eta = \frac{\sqrt{3}}{2} \xi\_1 \left( X - \frac{T}{3\xi\_1^2} \right) + a\_1. \tag{86}$$

Let us now consider the two-soliton solution of the VPE. In this case M Xð Þ ; T is a 4 � 4 matrix. We will not give the explicit form here, but we find that

$$\det M(\mathbf{X}, T) = \left(\mathbf{1} + q\_1^2 + q\_2^2 + b^2 q\_1^2 q\_2^2\right)^2,\tag{87}$$

where

The 2N � 2N matrix M Xð Þ ; T is defined as in relationship (6.36) in [34] by

� � � � �<sup>1</sup> � �

¼ iω3ξn, γ

¼ �iω2ξn, γ

For the N-soliton solution, there are N arbitrary constants ξ<sup>n</sup> and N arbitrary

The final result for the N-soliton solution of the VPE is defined by relationship

WXXT þ ð Þ 1 þ WT WX ¼ 0,

T

T

2 ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup>

2 ξ2 1sech<sup>2</sup>

Consequently, from Eq. (81) we have the one-soliton solution of the VPE

from (85) that β<sup>1</sup> is real. Note that with β1=ξ<sup>1</sup> <0 we have the real solution in the

we need first to calculate the 2 � 2 matrix M Xð Þ ; T according to (82) with N ¼ 1.

2ξ<sup>1</sup>

<sup>1</sup> � <sup>ω</sup>3β<sup>1</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup>

exp ffiffiffi 3 <sup>p</sup> <sup>ξ</sup><sup>1</sup> <sup>X</sup> � <sup>T</sup>

ffiffiffi 3 p

<sup>2</sup> <sup>ξ</sup><sup>1</sup> <sup>X</sup> � <sup>T</sup>

� � is an arbitrary constant. Since U is real, it follows

3ξ<sup>2</sup> 1

� �

� �

� � � � � � <sup>2</sup>

þ 3λ<sup>1</sup> ζ

λ<sup>j</sup> ζ ð Þk 1 � �

ð Þ k 1

� �

� λ<sup>1</sup> ζ ð Þl 1

ð Þ 2n�1

ð Þ 2n

T þ λ<sup>j</sup> ζ

<sup>12</sup> ¼ ω2βn, γ

<sup>12</sup> ¼ 0, γ

exp 2iω3ξ1<sup>X</sup> � ffiffiffi

3ξ<sup>2</sup> 1

þ α<sup>1</sup>

exp ffiffiffi 3 <sup>p</sup> <sup>ξ</sup>1<sup>X</sup> � ffiffiffi

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup> � ��<sup>1</sup>

� �

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup> � ��<sup>1</sup>

: (84)

, (85)

� �

T

1

CCCCA

T

(83)

ð Þk 1 � �

� λ<sup>1</sup> ζ ð Þl 1

> ð Þ 2n�1 <sup>13</sup> ¼ 0,

ð Þ 2n <sup>13</sup> ¼ ω3βn:

X

(82)

� � � �

� � ,

ð Þk 1 � � � � �<sup>1</sup>

exp � 3λ<sup>j</sup> ζ

ð Þ 2n�1 1 � �

ð Þ 2n 1 � �

6.1 Examples of one- and two-soliton solutions of the VPE

In order to obtain the one-soliton solution of the VPE (27)

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup> � ��<sup>1</sup>

3 <sup>p</sup> <sup>ξ</sup><sup>1</sup> � ��<sup>1</sup>

� �

det M Xð Þ¼ ; <sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>1</sup>

� � iω3β<sup>1</sup>

Mklð Þ¼ X; T δkl � ∑

n ¼ 1, 2, …,N,

ð Þ 2n�1 1 � �

constants βn.

(81) with (82).

We find that the matrix is

and its determinant is

where <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

30

exp ffiffiffi 3 <sup>p</sup> <sup>ξ</sup>1<sup>X</sup> � ffiffiffi

exp �2iω2ξ1<sup>X</sup> � ffiffiffi

U Xð Þ¼ ; <sup>T</sup> WXð Þ¼ <sup>X</sup>; <sup>T</sup> <sup>9</sup>

<sup>2</sup> ln <sup>β</sup>1=<sup>2</sup> ffiffiffi

form of the singular soliton [41].

<sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup>

<sup>1</sup> � <sup>ω</sup>2β<sup>1</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>1</sup>

0

BBBB@

�iω2β<sup>1</sup> 2ξ<sup>1</sup>

and

λ<sup>1</sup> ζ

λ<sup>1</sup> ζ ð Þ 2n 1 � �

3 j¼2 γ ð Þk <sup>1</sup><sup>j</sup> ð Þ 0

Research Advances in Chaos Theory

¼ iω2ξn, λ<sup>2</sup> ζ

¼ �iω3ξn, λ<sup>3</sup> ζ

$$q\_i = \exp\left[\frac{\sqrt{3}}{2}\xi\_i \left(X - \frac{T}{3\xi\_i^2}\right) + a\_i\right], \quad b^2 = \left(\frac{\xi\_2 - \xi\_1}{\xi\_2 + \xi\_1}\right)^2 \frac{\xi\_1^2 + \xi\_2^2 - \xi\_1\xi\_2}{\xi\_1^2 + \xi\_2^2 + \xi\_1\xi\_2}, \tag{88}$$

and <sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ln <sup>β</sup>i=<sup>2</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>ξ</sup><sup>i</sup> � � are arbitrary constants. The two-soliton solution to the VPE as found by the IST method is given by (81) together with (87).

Finally we note that comparison of (81) with W ¼ 6 lnð Þf <sup>X</sup> from (42) shows that

$$
\ln \left( \det M(X, T) \right) = 2 \ln \left( f \right). \tag{89}
$$

so that det M Xð Þ ; T is a perfect square for arbitrary N.

#### 6.2 The two-loop-like solitons of the VE

We discuss the two-loop soliton solution of the VE in more detail. Let us consider what happens in x-t space. The relations (20), (25) and (29) determine the solutions in x-t throughout the solutions in X-T. In these coordinates x-t, we have the loop-like solitons.

The shifts, δi, of the two-loop solitons u<sup>1</sup> and u<sup>2</sup> in the positive x-direction due to the interaction may be computed as follows. The larger loop soliton is always shifted forwards by the interaction. However, for smaller u<sup>2</sup> with r ¼ ξ1=ξ2, there is a value rc ¼ 0:88867 in that we have a different form of the phase shift:


At first sight it might seem that the behaviour in (b) and (c) contradicts conservation of 'momentum'. That this is not so is justified as follows. By integrating (9) with respect to x, we find that Ð <sup>∞</sup> �<sup>∞</sup> udx <sup>¼</sup> 0; also, by multiplying (9) by <sup>x</sup> and integrating with respect to x, we obtain Ð <sup>∞</sup> �<sup>∞</sup> xudx <sup>¼</sup> 0. Thus, in <sup>x</sup>-<sup>t</sup> space, the 'mass' of each soliton is zero, and 'momentum' is conserved whatever δ<sup>1</sup> and δ<sup>2</sup> may be. In particular δ<sup>1</sup> and δ<sup>2</sup> may have the same sign as in (c), or one of them may be zero as in (b).

Cases (a), (b) and (c) are illustrated in Figures 6–8, respectively; in these figures u is plotted against x for various values of t. For convenience in the figures, the interactions of solitons are shown in coordinates moving with speed ð Þ v<sup>1</sup> þ v<sup>2</sup> =2.

Figure 6.

The interaction process for two-loop solitons with ξ<sup>1</sup> ¼ 0.99 and ξ<sup>2</sup> ¼ 1 so that r ¼ 0.99 and δ<sup>1</sup> < 0.

7. Discussion on the loop-like solutions

7.1 Remarks on the existence and uniqueness theorem

solutions (Section 2).

Figure 8.

Loop-like Solitons

DOI: http://dx.doi.org/10.5772/intechopen.86583

restrictive conditions (see 7.2).

(see [43]).

33

We have already mentioned the important question on stability of loop-like

The interaction process for two-loop solitons with ξ<sup>1</sup> ¼ 0.5 and ξ<sup>2</sup> ¼ 1 so that r ¼ 0.5 and δ<sup>1</sup> > 0.

In [42], the existence and uniqueness theorem is formulated for system (one) differential equations. The loop-like solutions take place on travelling waves. In this case, the initial equation is reduced to an ordinary differential equation (ODE) (see Section 2). It has been this equation which we are exploring. Now we note some important remarks. In particular, in order to investigate the ODE (the solution on travelling waves), it is still necessary to reconcile this solution with the initial problem, which is described by the differential equation in partial derivatives (evolution equation). Consequently, the ambiguous solutions for the ODE during their reconstruction into the initial coordinates should be checked by means of some

It is necessary to note that if the conditions of the existence and uniqueness theorem break down, then nevertheless, this does not restrict the existence of solutions. Hence, the solutions can exist, for example, the multivalued solutions. Here we point out an example: the exact solutions for the Camassa-Holm equation (CHE) and the Degasperis-Procesi equation (DPE) can be constructed as the component solutions, through separate parts (branches) of solutions

The selection of possible multivalued solutions will be discussed in 7.2.

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

Figure 8. The interaction process for two-loop solitons with ξ<sup>1</sup> ¼ 0.5 and ξ<sup>2</sup> ¼ 1 so that r ¼ 0.5 and δ<sup>1</sup> > 0.

#### 7. Discussion on the loop-like solutions

We have already mentioned the important question on stability of loop-like solutions (Section 2).

#### 7.1 Remarks on the existence and uniqueness theorem

In [42], the existence and uniqueness theorem is formulated for system (one) differential equations. The loop-like solutions take place on travelling waves. In this case, the initial equation is reduced to an ordinary differential equation (ODE) (see Section 2). It has been this equation which we are exploring. Now we note some important remarks. In particular, in order to investigate the ODE (the solution on travelling waves), it is still necessary to reconcile this solution with the initial problem, which is described by the differential equation in partial derivatives (evolution equation). Consequently, the ambiguous solutions for the ODE during their reconstruction into the initial coordinates should be checked by means of some restrictive conditions (see 7.2).

It is necessary to note that if the conditions of the existence and uniqueness theorem break down, then nevertheless, this does not restrict the existence of solutions. Hence, the solutions can exist, for example, the multivalued solutions. Here we point out an example: the exact solutions for the Camassa-Holm equation (CHE) and the Degasperis-Procesi equation (DPE) can be constructed as the component solutions, through separate parts (branches) of solutions (see [43]).

The selection of possible multivalued solutions will be discussed in 7.2.

Figure 6.

Research Advances in Chaos Theory

Figure 7.

32

The interaction process for two-loop solitons with ξ<sup>1</sup> ¼ 0.99 and ξ<sup>2</sup> ¼ 1 so that r ¼ 0.99 and δ<sup>1</sup> < 0.

The interaction process for two-loop solitons with ξ<sup>1</sup> ¼ 0.88867 and ξ<sup>2</sup> ¼ 1 so that r ¼ 0.88867 and δ<sup>1</sup> = 0.

#### 7.2 Selection for the loop-like solutions

Solutions must satisfy the following conditions:

1. At the point <sup>η</sup> <sup>¼</sup> 0, the solution must pass over the ellipse <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>vη<sup>2</sup> ð Þηη <sup>¼</sup> <sup>0</sup> (see Eq. (4.3) in [3]);

unambiguous. What happens in our case for high frequency when the dissipative term has the form αu (see Eq. (18) in [29])? Will the inclusion of dissipation give

By direct integration of Eq. (8) (written in terms of the variables (11)) within the neighbourhood of singular points z ¼ 0 where z<sup>η</sup> ! �∞ and z<sup>τ</sup> ≪ zη, it can be derived (see [3]) that the dissipative term, with dissipation parameter less than some limit value α<sup>∗</sup> , does not destroy the loop-like solutions. Now we give a

Since the solution to the VE has a parametric form (15) and (16), there is a space of variables in which the solution is a single-valued function. Hence, we can solve the problem of the ambiguous solution. A number of states with their thermodynamic parameters can occupy one microvolume. It is assumed that the interaction between the separated states occupying one microvolume can be neglected in comparison with the interaction between the particles of one thermodynamic state. Even if we take into account the interaction between the separated states in accordance with the dynamic state equation (5), for high frequencies, a dissipative term arises which is similar to the corresponding term in Eq. (7) but with the other relaxation time. In this sense the separated terms are distributed in space, but describing the wave process, we consider them as interpenetratable. A similar situation, when several components with different hydrodynamic parameters occupy one microvolume, has been assumed in mixture theory (see, for instance, [48]). Such a fundamental assumption in the theory of mixtures is physically impossible (see [48], p.7), but it is appropriate in the sense that separated compo-

Consequently, the following three observations show that, in the framework of the approach considered here, there are multivalued solutions when we model high-frequency wave processes: (1) All parts of loop-like solution are stable to perturbations. (2) Dissipation does not destroy the loop-like solutions. (3) The investigation regarding the interaction of the solitons has shown that it is necessary to take into account the whole ambiguous solution and not just the

Loop-like solitons are a class of interesting wave phenomena, which take place in some nonlinear systems. This interest consisted not only in the interpretation of the solutions obtained but also in the explanation of the experimental results. The ambiguous structure of the loop-like solutions is similar to the loop soliton solution to an equation that models a stretched rope [44]. Loop-like solitons on a vortex filament were investigated by Hasimoto [45] and Lamb, Jr. [46]. The loop-like solutions appear in description of physical phenomena, in particular, electromagnetic terahertz pulses in asymmetric molecules [49], high-frequency perturbations

in a relaxation medium [3, 50, 51] and soliton in ferrites [52]. As a typical multivalued structure, loop soliton has been discussed in some possible physical

It must be admitted that we are a long way still from complete awareness of physical processes which can be described by loop-like solutions. However, the approach, considered here, will hopefully be interesting and useful in understanding the birth and death process for particles, since the mass and momentum of individual loop-like soliton are zero. Furthermore, the investigations in optics, magnetism and hydrodynamics clearly indicate the acceptability of the approach on loop-like solitons. Indeed, the phase shifts observed at interaction of solitons can be

fields including particle physics [53] and quantum field theory [54].

explained by means of loop-like solutions.

rise to unambiguous solutions?

DOI: http://dx.doi.org/10.5772/intechopen.86583

Loop-like Solitons

physical interpretation to ambiguous solutions.

nents are multi-velocity interpenetratable continua.

separate parts.

7.4 Conclusion

35


Thus, we cannot arbitrarily combine the solutions at η ¼ 0. The solutions, in particular, solitons should be specific loop-like form.

#### 7.3 Physical interpretation of the multivalued solutions

From the mathematical point of view, an ambiguous solution does not present difficulties, whereas the physical interpretation of ambiguity always presents some difficulties. In this connection the problem of ambiguous solutions is regarded as important. The problem consists in whether the ambiguity has a physical nature or is related to the incompleteness of the mathematical model, in particular to the lack of dissipation.

We will consider the problem related to the singular points when dissipation takes place. At these points the dissipative term α <sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>x</sup> tends to infinity. The question arises: Are there solutions of Eq. (8) in a loop-like form? That the dissipation is likely to destroy the loop-like solutions can be associated with the following wellknown fact [5]. For the simplest nonlinear equation without dispersion and without dissipation, namely,

$$
\frac{
\partial u
}{
\partial t
} + u \frac{
\partial u
}{
\partial \mathbf{x}
} = \mathbf{0},
\tag{90}
$$

any initial smooth solution with boundary conditions

$$\left.u\right|\_{\mathbf{x}\to+\infty} = \mathbf{0}, \qquad \left.u\right|\_{\mathbf{x}\to-\infty} = \mathfrak{u}\_{0} = \text{const.} > \mathbf{0}$$

becomes ambiguous in the final analysis. When dissipation is considered, we have the Burgers equation [47]:

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial \mathbf{x}} + \mu \frac{\partial^2 u}{\partial \mathbf{x}^2} = \mathbf{0}.$$

The dissipative term in this equation and in Eq. (6) for low frequency is coincident. The inclusion of the dissipative term transforms the solutions so that they cannot be ambiguous as a result of evolution. The wave parameters are always

7.2 Selection for the loop-like solutions

2. According to the conservation law Ð �<sup>∞</sup>

particular, solitons should be specific loop-like form.

takes place. At these points the dissipative term α <sup>∂</sup><sup>u</sup>

7.3 Physical interpretation of the multivalued solutions

(see Eq. (4.3) in [3]);

Research Advances in Chaos Theory

point 1 takes place.

point 1 if α ! 0.

satisfying point 2.

dissipation, namely,

34

have the Burgers equation [47]:

Solutions must satisfy the following conditions:

1. At the point <sup>η</sup> <sup>¼</sup> 0, the solution must pass over the ellipse <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>vη<sup>2</sup> ð Þηη <sup>¼</sup> <sup>0</sup>

'mass' of individual soliton equals to zero. This condition will be satisfied if

3. As you know [3], taking into account dissipation in the physical process allows one to select a solution from an array of possible solutions that are inherent to the equation without dissipation. This condition also selects a solution as in a

4.During the interaction of the solitons [24, 29], you must take into account all the parts of loop-like soliton (see end of Section 7.3). The soliton has a form

Thus, we cannot arbitrarily combine the solutions at η ¼ 0. The solutions, in

From the mathematical point of view, an ambiguous solution does not present difficulties, whereas the physical interpretation of ambiguity always presents some difficulties. In this connection the problem of ambiguous solutions is regarded as important. The problem consists in whether the ambiguity has a physical nature or is related to the incompleteness of the mathematical model, in particular to the lack of dissipation. We will consider the problem related to the singular points when dissipation

arises: Are there solutions of Eq. (8) in a loop-like form? That the dissipation is likely to destroy the loop-like solutions can be associated with the following wellknown fact [5]. For the simplest nonlinear equation without dispersion and without

<sup>u</sup>j j <sup>x</sup>!þ<sup>∞</sup> <sup>¼</sup> <sup>0</sup>; <sup>u</sup> <sup>x</sup>!�<sup>∞</sup> <sup>¼</sup> <sup>u</sup><sup>0</sup> <sup>¼</sup> const:<sup>&</sup>gt; <sup>0</sup>

becomes ambiguous in the final analysis. When dissipation is considered, we

The dissipative term in this equation and in Eq. (6) for low frequency is coincident. The inclusion of the dissipative term transforms the solutions so that they cannot be ambiguous as a result of evolution. The wave parameters are always

∂u ∂t þ u ∂u

any initial smooth solution with boundary conditions

∂u ∂t þ u ∂u ∂x þ μ ∂2 u <sup>∂</sup>x<sup>2</sup> <sup>¼</sup> <sup>0</sup>:

<sup>∞</sup> u xð Þ ; t dx ¼ const � 0 for t>0. The

<sup>∂</sup><sup>x</sup> tends to infinity. The question

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup>, (90)

unambiguous. What happens in our case for high frequency when the dissipative term has the form αu (see Eq. (18) in [29])? Will the inclusion of dissipation give rise to unambiguous solutions?

By direct integration of Eq. (8) (written in terms of the variables (11)) within the neighbourhood of singular points z ¼ 0 where z<sup>η</sup> ! �∞ and z<sup>τ</sup> ≪ zη, it can be derived (see [3]) that the dissipative term, with dissipation parameter less than some limit value α<sup>∗</sup> , does not destroy the loop-like solutions. Now we give a physical interpretation to ambiguous solutions.

Since the solution to the VE has a parametric form (15) and (16), there is a space of variables in which the solution is a single-valued function. Hence, we can solve the problem of the ambiguous solution. A number of states with their thermodynamic parameters can occupy one microvolume. It is assumed that the interaction between the separated states occupying one microvolume can be neglected in comparison with the interaction between the particles of one thermodynamic state. Even if we take into account the interaction between the separated states in accordance with the dynamic state equation (5), for high frequencies, a dissipative term arises which is similar to the corresponding term in Eq. (7) but with the other relaxation time. In this sense the separated terms are distributed in space, but describing the wave process, we consider them as interpenetratable. A similar situation, when several components with different hydrodynamic parameters occupy one microvolume, has been assumed in mixture theory (see, for instance, [48]). Such a fundamental assumption in the theory of mixtures is physically impossible (see [48], p.7), but it is appropriate in the sense that separated components are multi-velocity interpenetratable continua.

Consequently, the following three observations show that, in the framework of the approach considered here, there are multivalued solutions when we model high-frequency wave processes: (1) All parts of loop-like solution are stable to perturbations. (2) Dissipation does not destroy the loop-like solutions. (3) The investigation regarding the interaction of the solitons has shown that it is necessary to take into account the whole ambiguous solution and not just the separate parts.

#### 7.4 Conclusion

Loop-like solitons are a class of interesting wave phenomena, which take place in some nonlinear systems. This interest consisted not only in the interpretation of the solutions obtained but also in the explanation of the experimental results. The ambiguous structure of the loop-like solutions is similar to the loop soliton solution to an equation that models a stretched rope [44]. Loop-like solitons on a vortex filament were investigated by Hasimoto [45] and Lamb, Jr. [46]. The loop-like solutions appear in description of physical phenomena, in particular, electromagnetic terahertz pulses in asymmetric molecules [49], high-frequency perturbations in a relaxation medium [3, 50, 51] and soliton in ferrites [52]. As a typical multivalued structure, loop soliton has been discussed in some possible physical fields including particle physics [53] and quantum field theory [54].

It must be admitted that we are a long way still from complete awareness of physical processes which can be described by loop-like solutions. However, the approach, considered here, will hopefully be interesting and useful in understanding the birth and death process for particles, since the mass and momentum of individual loop-like soliton are zero. Furthermore, the investigations in optics, magnetism and hydrodynamics clearly indicate the acceptability of the approach on loop-like solitons. Indeed, the phase shifts observed at interaction of solitons can be explained by means of loop-like solutions.

Research Advances in Chaos Theory

### Author details

Vyacheslav O. Vakhnenko<sup>1</sup> \*, E. John Parkes<sup>2</sup> and Dmitri B. Vengrovich<sup>1</sup> References

Loop-like Solitons

0198400902021100

qj.49710444026

[2] Landau L, Lifshitz E. Fluids

[3] Vakhnenko V. High-frequency soliton-like waves in a relaxing medium. Journal of Mathematical Physics. 1999; 40:2011-2020. DOI: 10.1063/1.532847

[4] Gardner C, Greene J, Kruskal M, Miura R. Method for solving the Korteweg-deVries equation. Physical Review Letters. 1967;19:1095-1097. DOI:

[5] Dodd R, Eilbeck J, Gibbon J, Morris H. Solitons and Nonlinear Wave Equations. London: Academic Press;

[6] Su C, Gardner C. Korteweg-de Vries equation and generalizations. III. Derivation of Korteweg-de Vries equation and Burgers equation. Journal of Mathematical Physics. 1969;10: 536-539. DOI: 10.1063/1.1664873

[7] Ablowitz M, Segur H. Solitons and

Philadelphia: SIAM Press; 1981. 426 p.

[8] Novikov S, Manakov S, Pitaevskii L, Zakharov V. Theory of Solitons. The Inverse Scattering Methods. New York

Corporation; 1984. p. 286. ISBN 978-0-

[9] Hirota R. Direct methods in soliton theory. In: Bullough R, Caudrey P, editors. Solitons. New York—Berlin: Springer; 1980. pp. 157-176. DOI: 10.1007/978-3-642-81448-8

Inverse Scattering Transform.

DOI: 10.1137/1.9781611970883

—London: Plenum Publishing

306-10977-5

37

10.1103/PhysRevLett.19.1095

1982. p. 640. DOI: 10.1002/

zamm.19850650811

1978. p. 536. DOI: 10.1002/

[1] Clarke J. Lectures on plane waves in reacting gases. Annales de Physique. 1984;9:211-306. DOI: 10.1051/anphys:

DOI: http://dx.doi.org/10.5772/intechopen.86583

[10] Hirota R. The Direct Method in Soliton Theory. Cambridge: Cambridge University Press; 2004. 198 p. 10.1017/

[11] Newell A. Solitons in Mathematics and Physics. Philadelphia: SIAM; 1985.

Hamiltonian Methods in the Theory of Solitons. New York, Verlag: Berlin-Heidelberg, Springer; 1987. p. 592. DOI:

CBO9780511543043

p. 244. ISBN 0898711967

[12] Faddeev L, Takhtajan L.

10.1007/978-3-540-69969-9

[13] Wazwaz A. Partial Differential Equations and Solitary Waves Theory. Berlin-Heidelberg: Springer; 2009. p. 700. DOI: 10.1007/978-3-642-00251-9

[14] Vakhnenko V. Solitons in a nonlinear model medium. Journal of Physics A: Mathematical and General. 1992;25:4181-4187. DOI: 10.1088/

[15] Parkes J. The stability of solutions of Vakhnenko's equation. Journal of Physics A: Mathematical and General. 1993;26:6469-6475. DOI: 10.1088/

[16] Ostrovsky L. Nonlinear internal waves in a rotating ocean. Oceanology. 1978;18:119-125. https://ci.nii.ac.jp/naid/ 10010464167 ID. (NAID) 10010464167

[17] Byrd P, Friedman D. Handbook of Elliptic Integrals for Engineers and Scientists. 2nd ed. Berlin: Springer-Verlag; 1971. 276 p. DOI: 10.1007/978-3-

[18] Vakhnenko V, Parkes J. The two loop soliton solution of the Vakhnenko equation. Nonlinearity. 1998;11:1457- 1464. DOI: 10.1088/0951-7715/11/6/001

[19] Morrison A, Parkes J, Vakhnenko V. The N loop soliton solution of the

0305-4470/25/15/025

0305-4470/26/22/040

642-65138-0

Mechanics. New York: Pergamon Press;

1 Institute of Geophysics, Ukrainian Academy of Sciences, Kyiv, Ukraine

2 Department of Mathematics, University of Strathclyde, Glasgow, UK

\*Address all correspondence to: vakhnenko@ukr.net

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Clarke J. Lectures on plane waves in reacting gases. Annales de Physique. 1984;9:211-306. DOI: 10.1051/anphys: 0198400902021100

[2] Landau L, Lifshitz E. Fluids Mechanics. New York: Pergamon Press; 1978. p. 536. DOI: 10.1002/ qj.49710444026

[3] Vakhnenko V. High-frequency soliton-like waves in a relaxing medium. Journal of Mathematical Physics. 1999; 40:2011-2020. DOI: 10.1063/1.532847

[4] Gardner C, Greene J, Kruskal M, Miura R. Method for solving the Korteweg-deVries equation. Physical Review Letters. 1967;19:1095-1097. DOI: 10.1103/PhysRevLett.19.1095

[5] Dodd R, Eilbeck J, Gibbon J, Morris H. Solitons and Nonlinear Wave Equations. London: Academic Press; 1982. p. 640. DOI: 10.1002/ zamm.19850650811

[6] Su C, Gardner C. Korteweg-de Vries equation and generalizations. III. Derivation of Korteweg-de Vries equation and Burgers equation. Journal of Mathematical Physics. 1969;10: 536-539. DOI: 10.1063/1.1664873

[7] Ablowitz M, Segur H. Solitons and Inverse Scattering Transform. Philadelphia: SIAM Press; 1981. 426 p. DOI: 10.1137/1.9781611970883

[8] Novikov S, Manakov S, Pitaevskii L, Zakharov V. Theory of Solitons. The Inverse Scattering Methods. New York —London: Plenum Publishing Corporation; 1984. p. 286. ISBN 978-0- 306-10977-5

[9] Hirota R. Direct methods in soliton theory. In: Bullough R, Caudrey P, editors. Solitons. New York—Berlin: Springer; 1980. pp. 157-176. DOI: 10.1007/978-3-642-81448-8

[10] Hirota R. The Direct Method in Soliton Theory. Cambridge: Cambridge University Press; 2004. 198 p. 10.1017/ CBO9780511543043

[11] Newell A. Solitons in Mathematics and Physics. Philadelphia: SIAM; 1985. p. 244. ISBN 0898711967

[12] Faddeev L, Takhtajan L. Hamiltonian Methods in the Theory of Solitons. New York, Verlag: Berlin-Heidelberg, Springer; 1987. p. 592. DOI: 10.1007/978-3-540-69969-9

[13] Wazwaz A. Partial Differential Equations and Solitary Waves Theory. Berlin-Heidelberg: Springer; 2009. p. 700. DOI: 10.1007/978-3-642-00251-9

[14] Vakhnenko V. Solitons in a nonlinear model medium. Journal of Physics A: Mathematical and General. 1992;25:4181-4187. DOI: 10.1088/ 0305-4470/25/15/025

[15] Parkes J. The stability of solutions of Vakhnenko's equation. Journal of Physics A: Mathematical and General. 1993;26:6469-6475. DOI: 10.1088/ 0305-4470/26/22/040

[16] Ostrovsky L. Nonlinear internal waves in a rotating ocean. Oceanology. 1978;18:119-125. https://ci.nii.ac.jp/naid/ 10010464167 ID. (NAID) 10010464167

[17] Byrd P, Friedman D. Handbook of Elliptic Integrals for Engineers and Scientists. 2nd ed. Berlin: Springer-Verlag; 1971. 276 p. DOI: 10.1007/978-3- 642-65138-0

[18] Vakhnenko V, Parkes J. The two loop soliton solution of the Vakhnenko equation. Nonlinearity. 1998;11:1457- 1464. DOI: 10.1088/0951-7715/11/6/001

[19] Morrison A, Parkes J, Vakhnenko V. The N loop soliton solution of the

Author details

36

Vyacheslav O. Vakhnenko<sup>1</sup>

Research Advances in Chaos Theory

\*, E. John Parkes<sup>2</sup> and Dmitri B. Vengrovich<sup>1</sup>

1 Institute of Geophysics, Ukrainian Academy of Sciences, Kyiv, Ukraine

2 Department of Mathematics, University of Strathclyde, Glasgow, UK

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: vakhnenko@ukr.net

provided the original work is properly cited.

Vakhnenko equation. Nonlinearity. 1999;12:1427-1437. DOI: 10.1088/ 0951-7715/12/5/314

[20] Estévez P. Reciprocal transformations for a spectral problem in 2+1 dimensions. Theoretical and Mathematical Physics. 2009;159: 763-769. DOI: 10.4213/tmf6360

[21] Abazari R. Application of G<sup>0</sup> ð Þ =G expansion method to travelling wave solutions of three nonlinear evolution equation. Computers and Fluids. 2010; 39:1957-1963. DOI: 10.1016/j. compfluid.2010.06.024

[22] Majida F, Trikib H, Hayatc T, Aldossaryd O, Biswase A. Solitary wave solutions of the Vakhnenko-Parkes equation. Nonlinear Analysis: Modelling and Control. 2012;17:60-66 https:// www.mii.lt/na/issues/NA\_1701/ NA17105.pd

[23] Ye Y, Song J, Shen S, Di Y. New coherent structures of the Vakhnenko-Parkes equation. Results in Physics. 2012;2:170-174. DOI: 10.1016/j. rinp.2012.09.011

[24] Vakhnenko V, Parkes J. The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos, Solitons and Fractals. 2002;13:1819-1826. DOI: 10.1016/S0960-0779(01)00200-4

[25] Vakhnenko V, Parkes J. The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method. Chaos, Solitons and Fractals. 2012;45:846-852. DOI: 10.1016/ j.chaos.2012.02.019

[26] Vakhnenko V, Parkes J. Solutions associated with discrete and continuous spectrums in the inverse scattering method for the Vakhnenko-Parkes equation. Progress in Theoretical Physics. 2012;127:593-613. DOI: 10.1143/ PTP.127.593

[27] Vakhnenko V, Parkes J. Special singularity function for continuous part of the spectral data in the associated eigenvalue problem for nonlinear equations. Journal of Mathematical Physics. 2012;53:063504. DOI: 10.1063/ 1.4726168

[35] Caudrey P. The inverse problem

DOI: http://dx.doi.org/10.5772/intechopen.86583

Letters A. 1980;79:264-268. DOI: 10.1016/0375-9601(80)90343-6

[36] Zakharov V. On stochastization of one-dimensional chains of nonlinear oscillators. Soviet Physics - JETP. 1974;38: 108-110 http://cds.cern.ch/record/407338

[37] Deift P, Tomei C, Trubowitz E. Inverse scattering and the Boussinesq equation. Commun. Pure and Applied Mathematics Journal. 1982;35:567-628.

[38] Hirota R, Satsuma J. A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice. Progress of Theoretical Physics Supplement. 1976;59:64-100.

[39] Musette M, Conte R. Algorithmic method for deriving lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations. Journal of

[40] Clarkson P, Mansfield E. Symmetry reductions and exact solutions of shallow water wave equations. Acta Applicandae Mathematicae. 1995;39: 245-276. DOI: 10.1007/BF00994636

[41] Wazwaz A. N-soliton solutions for the Vakhnenko equation and its generalized forms. Physica Scripta. 2010;82:065006(7). DOI: 10.1088/

[42] Pontryagin L. Ordinary Differential Equations. London: Addison-Wesley Publishing Company; 1962. p. 298. DOI:

[43] Lenells J. Traveling wave solutions of the Camassa–holm equation. Journal of differential equation. 2005;217:393-430.

0031-8949/82/06/065006

10.1002/zamm.19630430924

DOI: 10.1016/j.jde.2004.09.007

39

Mathematical Physics. 1991;32: 1450-1457. DOI: 10.1063/1.529302

DOI: 10.1002/cpa.3160350502

DOI: 10.1143/PTPS.59.64

u. Physics

[44] Konno K, Ichikawa Y, Wadati M. A

[45] Hasimoto H. A soliton on a vortex filament. Journal of Fluid Mechanics. 1972;51:477-485. DOI: 10.1017/

[46] Lamb G Jr. Solitons on moving space curves. Journal of Mathematical Physics. 1977;18:1654-1661. DOI:

[47] Burgers J. A mathematical model illustrating the theory of turbulence. In: von Mises R, von Kármán T, editors. Advances in Applied Mechanics. New York: Academic Press Inc.; 1948. pp. 171-199. DOI: 10.1016/S0065-2156

[48] Rajagopal K, Tao L. Mechanics of Mixtures. Singapore: World Scientific

[49] Sazonov S, Ustinov N. Nonlinear propagation of the vector too short pulses in the medium with symmetric and asymmetric molecules. Journal of Experimental and Theoretical Physics. 2017;124:249-269. DOI: 10.7868/

[50] Kraenkel R, Leblond H, Manna M. An integrable evolution equation for surface waves in deep water. Journal of

Theoretical. 2014;47:025208(17). DOI: 10.1088/1751-8113/47/2/025208

[51] Kuetche V. Barothropic relaxing media under pressure perturbations: Nonlinear dynamics. Dynamics of Atmospheres and Oceans. 2015;72:21-37. DOI: 10.1016/j.dynatmoce.2015.10.001

[52] Kuetche V. Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions. Journal

Physics A: Mathematical and

Publishing; 1995. p. 195. ISBN

loop soliton propagating along a stretched rope. Journal of the Physical Society of Japan. 1981;50:1025-1026.

DOI: 10.1143/JPSJ.50.1025

S0022112072002307

10.1063/1.523453

(08)70100-5

9810215851

S0044451017020043

for the third order equation uxxx <sup>þ</sup> q xð Þux <sup>þ</sup> r xð Þ<sup>u</sup> ¼ �iζ<sup>3</sup>

Loop-like Solitons

[28] Vakhnenko V, Parkes J. The inverse problem for some special spectral data. Chaos, Solitons and Fractals. 2016;82: 116-124. DOI: 10.1016/j. chaos.2015.11.012

[29] Vakhnenko V, Parkes J. Approach in theory of nonlinear evolution equations: The Vakhnenko-Parkes equation. Advances in Mathematical Physics. 2016;2016:1-39. DOI: 10.1155/2016/ 2916582. Article ID 2916582

[30] Vakhnenko V, Parkes J, Morrison A. A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Solitons and Fractals. 2003;17: 683-692. DOI: 10.1016/S0960-0779(02) 00483-6

[31] Hirota R. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Progress in Theoretical Physics. 1974;52:1498-1512. DOI: 10.1143/PTP.52.1498

[32] Satsuma J, Kaup D. A Bäcklund transformation for a higher order Korteweg-de Vries equation. Journal of the Physical Society of Japan. 1977;43: 692-697. DOI: 10.1143/JPSJ.43.692

[33] Kaup D. On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx þ 6Qψ<sup>x</sup> þ 6Rψ ¼ λψ. Studies in Applied Mathematics. 1980; 62:189-216. DOI: 10.1002/ sapm1980623189

[34] Caudrey P. The inverse problem for a general N � N spectral equation. Physica D. 1982;6:51-66. DOI: 10.1016/ 0167-2789(82)90004-5

Loop-like Solitons DOI: http://dx.doi.org/10.5772/intechopen.86583

[35] Caudrey P. The inverse problem for the third order equation uxxx <sup>þ</sup> q xð Þux <sup>þ</sup> r xð Þ<sup>u</sup> ¼ �iζ<sup>3</sup> u. Physics Letters A. 1980;79:264-268. DOI: 10.1016/0375-9601(80)90343-6

Vakhnenko equation. Nonlinearity. 1999;12:1427-1437. DOI: 10.1088/

Research Advances in Chaos Theory

[27] Vakhnenko V, Parkes J. Special singularity function for continuous part of the spectral data in the associated eigenvalue problem for nonlinear equations. Journal of Mathematical Physics. 2012;53:063504. DOI: 10.1063/

[28] Vakhnenko V, Parkes J. The inverse problem for some special spectral data. Chaos, Solitons and Fractals. 2016;82:

[29] Vakhnenko V, Parkes J. Approach in theory of nonlinear evolution equations: The Vakhnenko-Parkes equation. Advances in Mathematical Physics. 2016;2016:1-39. DOI: 10.1155/2016/

[30] Vakhnenko V, Parkes J, Morrison A. A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Solitons and Fractals. 2003;17: 683-692. DOI: 10.1016/S0960-0779(02)

[31] Hirota R. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Progress in Theoretical Physics. 1974;52:1498-1512.

[32] Satsuma J, Kaup D. A Bäcklund transformation for a higher order Korteweg-de Vries equation. Journal of the Physical Society of Japan. 1977;43: 692-697. DOI: 10.1143/JPSJ.43.692

[33] Kaup D. On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx þ 6Qψ<sup>x</sup> þ 6Rψ ¼ λψ. Studies in Applied Mathematics. 1980;

[34] Caudrey P. The inverse problem for a general N � N spectral equation. Physica D. 1982;6:51-66. DOI: 10.1016/

DOI: 10.1143/PTP.52.1498

62:189-216. DOI: 10.1002/

0167-2789(82)90004-5

sapm1980623189

116-124. DOI: 10.1016/j. chaos.2015.11.012

2916582. Article ID 2916582

1.4726168

00483-6

transformations for a spectral problem in 2+1 dimensions. Theoretical and Mathematical Physics. 2009;159: 763-769. DOI: 10.4213/tmf6360

[21] Abazari R. Application of G<sup>0</sup> ð Þ =G expansion method to travelling wave solutions of three nonlinear evolution equation. Computers and Fluids. 2010;

39:1957-1963. DOI: 10.1016/j. compfluid.2010.06.024

NA17105.pd

rinp.2012.09.011

j.chaos.2012.02.019

PTP.127.593

38

[22] Majida F, Trikib H, Hayatc T, Aldossaryd O, Biswase A. Solitary wave solutions of the Vakhnenko-Parkes equation. Nonlinear Analysis: Modelling and Control. 2012;17:60-66 https:// www.mii.lt/na/issues/NA\_1701/

[23] Ye Y, Song J, Shen S, Di Y. New coherent structures of the Vakhnenko-Parkes equation. Results in Physics. 2012;2:170-174. DOI: 10.1016/j.

[24] Vakhnenko V, Parkes J. The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos, Solitons and Fractals. 2002;13:1819-1826. DOI: 10.1016/S0960-0779(01)00200-4

[25] Vakhnenko V, Parkes J. The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method. Chaos, Solitons and Fractals. 2012;45:846-852. DOI: 10.1016/

[26] Vakhnenko V, Parkes J. Solutions associated with discrete and continuous spectrums in the inverse scattering method for the Vakhnenko-Parkes equation. Progress in Theoretical Physics. 2012;127:593-613. DOI: 10.1143/

0951-7715/12/5/314

[20] Estévez P. Reciprocal

[36] Zakharov V. On stochastization of one-dimensional chains of nonlinear oscillators. Soviet Physics - JETP. 1974;38: 108-110 http://cds.cern.ch/record/407338

[37] Deift P, Tomei C, Trubowitz E. Inverse scattering and the Boussinesq equation. Commun. Pure and Applied Mathematics Journal. 1982;35:567-628. DOI: 10.1002/cpa.3160350502

[38] Hirota R, Satsuma J. A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice. Progress of Theoretical Physics Supplement. 1976;59:64-100. DOI: 10.1143/PTPS.59.64

[39] Musette M, Conte R. Algorithmic method for deriving lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations. Journal of Mathematical Physics. 1991;32: 1450-1457. DOI: 10.1063/1.529302

[40] Clarkson P, Mansfield E. Symmetry reductions and exact solutions of shallow water wave equations. Acta Applicandae Mathematicae. 1995;39: 245-276. DOI: 10.1007/BF00994636

[41] Wazwaz A. N-soliton solutions for the Vakhnenko equation and its generalized forms. Physica Scripta. 2010;82:065006(7). DOI: 10.1088/ 0031-8949/82/06/065006

[42] Pontryagin L. Ordinary Differential Equations. London: Addison-Wesley Publishing Company; 1962. p. 298. DOI: 10.1002/zamm.19630430924

[43] Lenells J. Traveling wave solutions of the Camassa–holm equation. Journal of differential equation. 2005;217:393-430. DOI: 10.1016/j.jde.2004.09.007

[44] Konno K, Ichikawa Y, Wadati M. A loop soliton propagating along a stretched rope. Journal of the Physical Society of Japan. 1981;50:1025-1026. DOI: 10.1143/JPSJ.50.1025

[45] Hasimoto H. A soliton on a vortex filament. Journal of Fluid Mechanics. 1972;51:477-485. DOI: 10.1017/ S0022112072002307

[46] Lamb G Jr. Solitons on moving space curves. Journal of Mathematical Physics. 1977;18:1654-1661. DOI: 10.1063/1.523453

[47] Burgers J. A mathematical model illustrating the theory of turbulence. In: von Mises R, von Kármán T, editors. Advances in Applied Mechanics. New York: Academic Press Inc.; 1948. pp. 171-199. DOI: 10.1016/S0065-2156 (08)70100-5

[48] Rajagopal K, Tao L. Mechanics of Mixtures. Singapore: World Scientific Publishing; 1995. p. 195. ISBN 9810215851

[49] Sazonov S, Ustinov N. Nonlinear propagation of the vector too short pulses in the medium with symmetric and asymmetric molecules. Journal of Experimental and Theoretical Physics. 2017;124:249-269. DOI: 10.7868/ S0044451017020043

[50] Kraenkel R, Leblond H, Manna M. An integrable evolution equation for surface waves in deep water. Journal of Physics A: Mathematical and Theoretical. 2014;47:025208(17). DOI: 10.1088/1751-8113/47/2/025208

[51] Kuetche V. Barothropic relaxing media under pressure perturbations: Nonlinear dynamics. Dynamics of Atmospheres and Oceans. 2015;72:21-37. DOI: 10.1016/j.dynatmoce.2015.10.001

[52] Kuetche V. Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions. Journal of Magnetism and Magnetic Materials. 2016;398:70-81. DOI: 10.1016/j. jmmm.2015.08.120

Chapter 3

Abstract

1. Introduction

41

founded in the process of repeated game.

Complex Dynamical Behavior of a

Bounded Rational Duopoly Game

In this chapter, we assume that two bounded rational firms not only pursue profit maximization but also take consumer surplus into account, so the objections of all the firms are combination of their profits and the consumer surplus. And then a dynamical duopoly Cournot model with bounded rationality is established. The existence and stability of the boundary equilibrium points and the Nash equilibrium of the model are discussed, respectively. And then the stability condition of the Nash equilibrium is given. The complex dynamical behavior of the system varies with parameters in the parameter space is studied by using the so-called 2D bifurcation diagram. The coexistence of multiple attractors is discussed through analyzing basins of attraction. It is found that not only two attractors can coexist, but also three or even four attractors may coexist in the established model. Then, the topological structure of basins of attraction and the global dynamics of the system are discussed through invertible map, critical curve and transverse Lyapunov exponent.

At last, the synchronization phenomenon of the built model is studied.

Keywords: bifurcation, chaos, duopoly, consumer surplus, synchronization

Oligopoly is a market between perfect monopoly and perfect competition [1]. With the application of chaos theory and nonlinear dynamic system into oligopoly models, the static game evolves into a dynamic game. Especially in recent years, with the rapid development of computer technology, a powerful tool has been provided for dealing with the complex nonlinear problems. And hence, the economists and the mathematicians can simulate the complex dynamical behavior of oligopoly market by using computer technology. Recently, a large number of scholars have improved the oligopoly models, and introduced bounded rationality (see [2, 3]), incomplete information [4], time delay [5], market entering and entering barriers [6], differentiated products [7] and other factors [8, 9] into the classical oligopoly models, and the bifurcation and chaos phenomenon were

However, all of the above discussions are mainly based on private enterprises, which pursuit the maximization of their own profits. In fact, the public ownership enterprises, which always aim at maximizing social welfare, and mixed ownership enterprises, which always aim at maximizing the weighted average of the social

with Consumer Surplus

Wei Zhou and Tong Chu

[53] Schleif M, Wunsch R. Thermodynamic properties of the SU(2) (f) chiral quark-loop soliton. European Physical Journal A: Hadrons and Nuclei. 1998;1:171-186. DOI: 10.1007/ s100500050046

[54] Matsutani S. The relation of lemniscate and a loop soliton as 3/2 and 1 spin fields along the modified Korteweg-de Vries equation. Modern Physics Letters A. 1995;10:717-721. DOI: 10.1142/S0217732395000764

#### Chapter 3

of Magnetism and Magnetic Materials. 2016;398:70-81. DOI: 10.1016/j.

Research Advances in Chaos Theory

Thermodynamic properties of the SU(2) (f) chiral quark-loop soliton. European Physical Journal A: Hadrons and Nuclei.

lemniscate and a loop soliton as 3/2 and

jmmm.2015.08.120

s100500050046

40

[53] Schleif M, Wunsch R.

1998;1:171-186. DOI: 10.1007/

[54] Matsutani S. The relation of

1 spin fields along the modified Korteweg-de Vries equation. Modern Physics Letters A. 1995;10:717-721. DOI:

10.1142/S0217732395000764

## Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

Wei Zhou and Tong Chu

#### Abstract

In this chapter, we assume that two bounded rational firms not only pursue profit maximization but also take consumer surplus into account, so the objections of all the firms are combination of their profits and the consumer surplus. And then a dynamical duopoly Cournot model with bounded rationality is established. The existence and stability of the boundary equilibrium points and the Nash equilibrium of the model are discussed, respectively. And then the stability condition of the Nash equilibrium is given. The complex dynamical behavior of the system varies with parameters in the parameter space is studied by using the so-called 2D bifurcation diagram. The coexistence of multiple attractors is discussed through analyzing basins of attraction. It is found that not only two attractors can coexist, but also three or even four attractors may coexist in the established model. Then, the topological structure of basins of attraction and the global dynamics of the system are discussed through invertible map, critical curve and transverse Lyapunov exponent. At last, the synchronization phenomenon of the built model is studied.

Keywords: bifurcation, chaos, duopoly, consumer surplus, synchronization

#### 1. Introduction

Oligopoly is a market between perfect monopoly and perfect competition [1]. With the application of chaos theory and nonlinear dynamic system into oligopoly models, the static game evolves into a dynamic game. Especially in recent years, with the rapid development of computer technology, a powerful tool has been provided for dealing with the complex nonlinear problems. And hence, the economists and the mathematicians can simulate the complex dynamical behavior of oligopoly market by using computer technology. Recently, a large number of scholars have improved the oligopoly models, and introduced bounded rationality (see [2, 3]), incomplete information [4], time delay [5], market entering and entering barriers [6], differentiated products [7] and other factors [8, 9] into the classical oligopoly models, and the bifurcation and chaos phenomenon were founded in the process of repeated game.

However, all of the above discussions are mainly based on private enterprises, which pursuit the maximization of their own profits. In fact, the public ownership enterprises, which always aim at maximizing social welfare, and mixed ownership enterprises, which always aim at maximizing the weighted average of the social

welfare and their own profits, are also widespread in the real economic environment. De Fraja and Delbono [10] found that the social welfare might be higher when a public ownership enterprise is a profit-maximizer rather than a social-welfaremaximizer. Matsumura [11] proposed that the social welfare could be improved through partial privatization of public enterprises. The research of Fujiwara [12] suggested that partial privatized public enterprises are more efficient than private enterprises. Elsadany and Awad [13] explored the complex dynamical behavior of competition between two partial public enterprises under the assumption of bounded rationality. However, the global dynamical behavior and synchronization behavior of semi-public enterprises, which corporate social responsibility into their objectives, are rarely studied. In this chapter, the occurrence of synchronization, the coexistence of attractors and the global dynamic of a duopoly game corporate consumer surplus are mainly discussed.

empirical studies have shown how the introduced corporate social responsibility affects firm's performance, where we interpret corporate social responsibility as either consumer surplus (for short CS) or social welfare (for short SW). In this chapter we take CS into account to analyze which firms have an incentive to exhibit corporate social responsibility as a means of maximizing their profits in a Cournot competition. Based on the above assumptions and the definition of consumer sur-

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

CS ¼

where p ∈ð Þ p; a is the price variable.

DOI: http://dx.doi.org/10.5772/intechopen.87200

objection function of firm i can be given by,

Oi ¼ ð Þ 1 � α<sup>i</sup> a � b q<sup>1</sup> þ q<sup>2</sup>

ða p

� � � <sup>c</sup> � �qi <sup>þ</sup>

¼ ð Þ 1 � α<sup>i</sup> ð Þþ a � c ð Þ 3α<sup>i</sup> � 2 bqi þ ð Þ 2α<sup>i</sup> � 1 bqj

qi

increase its output at period <sup>t</sup> <sup>þ</sup> 1, if <sup>∂</sup>Oið Þ<sup>t</sup>

∂qi

ð Þ¼ t þ 1 qi

where vi > 0, i ¼ 1, 2 is an adjustment parameter of firm i. The firm i will

<sup>T</sup> : <sup>q</sup>1ð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>1</sup> <sup>q</sup>1ð Þþ<sup>t</sup> <sup>v</sup>1q1ð Þ<sup>t</sup> ð Þ <sup>1</sup> � <sup>α</sup><sup>1</sup> ð Þþ <sup>a</sup> � <sup>c</sup> ð Þ <sup>3</sup>bα<sup>1</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup>1ðÞþ<sup>t</sup> ð Þ <sup>2</sup>bα<sup>1</sup> � <sup>b</sup> <sup>q</sup>2ð Þ<sup>t</sup> � � <sup>q</sup>2ð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>1</sup> <sup>q</sup>2ð Þþ<sup>t</sup> <sup>v</sup>2q2ð Þ<sup>t</sup> ð Þ <sup>1</sup> � <sup>α</sup><sup>2</sup> ð Þþ <sup>a</sup> � <sup>c</sup> ð Þ <sup>3</sup>bα<sup>2</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup>2ðÞþ<sup>t</sup> ð Þ <sup>2</sup>bα<sup>2</sup> � <sup>b</sup> <sup>q</sup>1ð Þ<sup>t</sup> � � (

Since the output of a firm cannot be negative, the initial conditions of map T

∂qi

a � p

<sup>b</sup> dp <sup>¼</sup> b q<sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

According to the above assumptions, the objective function of the firm i can be

where α<sup>i</sup> represents the weight of the consumer surplus in the objective function

It is now significant to specify the information set of both players regarding the objection functions, to determine the behaviors of the players with the change of time. We assume a discrete time ð Þ t∈ Z<sup>þ</sup> dynamic setting, where two firms with bounded rationality make decisions at the same time. That is, all firms do not have complete knowledge of their competitors' decisions and the market demands. So it can only use the local estimation of the steepest slope of the objection function at period t to determine the output at period t þ 1. By following Bischi et al. [16] and Fanti et al. [17], the adjustment mechanism of quantities with the change of time of

ðÞþt viqi

ð Þt ∂Oi ∂qi

ð Þ<sup>t</sup> <sup>&</sup>gt; 0. By substituting (8) into (9), we can get a two-dimensional

ð Þ<sup>t</sup> <sup>&</sup>gt;0. But the firm <sup>i</sup> will reduce its output at

1 2

of firm i, and 0≤ α<sup>i</sup> ≤1 always holds. By substituting (4) and (5) into (6), the

And the first-order condition of the objection function (7) is given as,

� �<sup>2</sup>

Oi ¼ ð Þ 1 � α<sup>i</sup> π<sup>i</sup> þ αiCS, 0≤ α<sup>i</sup> ≤1, i ¼ 1, 2 (6)

αib q<sup>1</sup> þ q<sup>2</sup> � �<sup>2</sup>

<sup>2</sup> (5)

, i ¼ 1, 2 (7)

, i, j ¼ 1, 2, i 6¼ j (8)

(9)

(10)

plus, CS can be written as,

given as,

∂Oi ∂qi

firm i can be obtained as,

period <sup>t</sup> <sup>þ</sup> 1, if <sup>∂</sup>Oið Þ<sup>t</sup>

map as,

belong to

43

#### 2. The model

Considering a duopolistic market where two firms produce homogeneous goods. In order to study the long-term behaviors of the duopoly market with quantity competition, we briefly present the economic setup leading to the final model in this chapter. The price and quantity of product of firm i are given by pi and qi respectively, with i ¼ 1, 2. We also assume the existence of a continuum of identical consumers which have preferences toward q<sup>1</sup> and q2.

Following Dixit [14] and Singh and Vives [15], we suppose that the utility function used in this chapter is quadratic and can be given by,

$$U(q\_1, q\_2) = a(q\_1 + q\_2) - \frac{b}{2} \left( q\_1^2 + 2q\_1 q\_2 + q\_2^2 \right) \tag{1}$$

where q1, q<sup>2</sup> are the quantity of goods produced by firm 1 and firm 2, respectively. a>0 represents the maximum price of a unit's commodity, b>0 represents the amount of its price decreases when the price of the product increases by one unit.

Suppose that the budget constraint of consumer is,

$$p\_1q\_1 + p\_2q\_2 = M \tag{2}$$

where p<sup>1</sup> and p<sup>2</sup> denote the prices of goods produced by firm 1 and firm 2, respectively. And M denotes the budget of the consumers on the product. The utility function of consumers is maximized under the budget constraint, and then the inverse demand function of the two firms can be obtained as,

$$p\_i = a - b(q\_1 + q\_2), i = 1, 2\tag{3}$$

This chapter discusses homogenous products, so here it is assumed that all these two players have the identical marginal cost. Therefore, the cost function of firm 1 and firm 2 are same and can be given by, C qð Þ¼ cq, where c>0 denotes the marginal cost of the goods and a>c always holds. Then, the profits of firm i, i ¼ 1, 2 can be obtained as follows,

$$\boldsymbol{a}\pi\_{i} = \left[\boldsymbol{a} - \boldsymbol{b}\left(\boldsymbol{q}\_{1} + \boldsymbol{q}\_{2}\right) - \boldsymbol{c}\right] \boldsymbol{q}\_{i}, i = \textbf{1, 2} \tag{4}$$

In the real market, there are a lot of firms, who not only pursue their own profits but also take corporate social responsibility into account. A large number of

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

empirical studies have shown how the introduced corporate social responsibility affects firm's performance, where we interpret corporate social responsibility as either consumer surplus (for short CS) or social welfare (for short SW). In this chapter we take CS into account to analyze which firms have an incentive to exhibit corporate social responsibility as a means of maximizing their profits in a Cournot competition. Based on the above assumptions and the definition of consumer surplus, CS can be written as,

$$\text{CS} = \int\_{p}^{a} \frac{a - \overline{p}}{b} d\overline{p} = \frac{b\left(q\_1 + q\_2\right)^2}{2} \tag{5}$$

where p ∈ð Þ p; a is the price variable.

welfare and their own profits, are also widespread in the real economic environment. De Fraja and Delbono [10] found that the social welfare might be higher when a public ownership enterprise is a profit-maximizer rather than a social-welfaremaximizer. Matsumura [11] proposed that the social welfare could be improved through partial privatization of public enterprises. The research of Fujiwara [12] suggested that partial privatized public enterprises are more efficient than private enterprises. Elsadany and Awad [13] explored the complex dynamical behavior of competition between two partial public enterprises under the assumption of bounded rationality. However, the global dynamical behavior and synchronization behavior of semi-public enterprises, which corporate social responsibility into their objectives, are rarely studied. In this chapter, the occurrence of synchronization, the coexistence of attractors and the global dynamic of a duopoly game corporate

Considering a duopolistic market where two firms produce homogeneous goods.

In order to study the long-term behaviors of the duopoly market with quantity competition, we briefly present the economic setup leading to the final model in this chapter. The price and quantity of product of firm i are given by pi and qi respectively, with i ¼ 1, 2. We also assume the existence of a continuum of identical

Following Dixit [14] and Singh and Vives [15], we suppose that the utility

� <sup>b</sup>

a>0 represents the maximum price of a unit's commodity, b>0 represents the amount of its price decreases when the price of the product increases by one unit.

where p<sup>1</sup> and p<sup>2</sup> denote the prices of goods produced by firm 1 and firm 2, respectively. And M denotes the budget of the consumers on the product. The utility function of consumers is maximized under the budget constraint, and then

This chapter discusses homogenous products, so here it is assumed that all these two players have the identical marginal cost. Therefore, the cost function of firm 1 and firm 2 are same and can be given by, C qð Þ¼ cq, where c>0 denotes the marginal cost of the goods and a>c always holds. Then, the profits of firm i, i ¼ 1, 2 can be

� <sup>c</sup> qi

but also take corporate social responsibility into account. A large number of

In the real market, there are a lot of firms, who not only pursue their own profits

<sup>2</sup> <sup>q</sup><sup>2</sup>

where q1, q<sup>2</sup> are the quantity of goods produced by firm 1 and firm 2, respectively.

<sup>1</sup> <sup>þ</sup> <sup>2</sup>q1q<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

p1q<sup>1</sup> þ p2q<sup>2</sup> ¼ M (2)

, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup> (3)

, i ¼ 1, 2 (4)

2 (1)

consumer surplus are mainly discussed.

Research Advances in Chaos Theory

consumers which have preferences toward q<sup>1</sup> and q2.

U q1; q<sup>2</sup>

function used in this chapter is quadratic and can be given by,

<sup>¼</sup> a q<sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

Suppose that the budget constraint of consumer is,

the inverse demand function of the two firms can be obtained as,

π<sup>i</sup> ¼ a � b q<sup>1</sup> þ q<sup>2</sup>

pi ¼ a � b q<sup>1</sup> þ q<sup>2</sup>

2. The model

obtained as follows,

42

According to the above assumptions, the objective function of the firm i can be given as,

$$\mathbf{O}\_{i} = (\mathbf{1} - a\_{i})\boldsymbol{\pi}\_{i} + a\_{i}\mathbf{C}\mathbf{S}, \quad \mathbf{0} \le a\_{i} \le \mathbf{1}, \ i = \mathbf{1}, \mathbf{2} \tag{6}$$

where α<sup>i</sup> represents the weight of the consumer surplus in the objective function of firm i, and 0≤ α<sup>i</sup> ≤1 always holds. By substituting (4) and (5) into (6), the objection function of firm i can be given by,

$$O\_i = (1 - a\_i) \left[ a - b \left( q\_1 + q\_2 \right) - c \right] q\_i + \frac{1}{2} a\_i b \left( q\_1 + q\_2 \right)^2, \quad i = 1, 2 \tag{7}$$

And the first-order condition of the objection function (7) is given as,

$$\frac{\partial O\_i}{\partial q\_i} = (\mathbf{1} - a\_i)(\mathbf{a} - c) + (\mathbf{3}a\_i - \mathbf{2})bq\_i + (\mathbf{2}a\_i - \mathbf{1})bq\_j, i, j = \mathbf{1}, \mathbf{2}, i \neq j \tag{8}$$

It is now significant to specify the information set of both players regarding the objection functions, to determine the behaviors of the players with the change of time. We assume a discrete time ð Þ t∈ Z<sup>þ</sup> dynamic setting, where two firms with bounded rationality make decisions at the same time. That is, all firms do not have complete knowledge of their competitors' decisions and the market demands. So it can only use the local estimation of the steepest slope of the objection function at period t to determine the output at period t þ 1. By following Bischi et al. [16] and Fanti et al. [17], the adjustment mechanism of quantities with the change of time of firm i can be obtained as,

$$q\_i(t+1) = q\_i(t) + \nu\_i q\_i(t) \frac{\partial O\_i}{\partial q\_i} \tag{9}$$

where vi > 0, i ¼ 1, 2 is an adjustment parameter of firm i. The firm i will increase its output at period <sup>t</sup> <sup>þ</sup> 1, if <sup>∂</sup>Oið Þ<sup>t</sup> ∂qi ð Þ<sup>t</sup> <sup>&</sup>gt;0. But the firm <sup>i</sup> will reduce its output at period <sup>t</sup> <sup>þ</sup> 1, if <sup>∂</sup>Oið Þ<sup>t</sup> ∂qi ð Þ<sup>t</sup> <sup>&</sup>gt; 0. By substituting (8) into (9), we can get a two-dimensional map as,

$$T: \begin{cases} q\_1(t+1) = q\_1(t) + v\_1 q\_1(t) \left[ (1-a\_1)(a-c) + (3ba\_1 - 2b)q\_1(t) + (2ba\_1 - b)q\_2(t) \right] \\ q\_2(t+1) = q\_2(t) + v\_2 q\_2(t) \left[ (1-a\_2)(a-c) + (3ba\_2 - 2b)q\_2(t) + (2ba\_2 - b)q\_1(t) \right] \end{cases} \tag{10}$$

Since the output of a firm cannot be negative, the initial conditions of map T belong to

$$F = \left\{ (q\_1, q\_2) : q\_1 \ge 0, q\_2 \ge 0, q\_1 + q\_2 \ne 0 \right\}.$$

Proof. By substituting the equilibrium E<sup>1</sup> into (11), the Jacobian matrix of map

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

<sup>3</sup><sup>α</sup> � <sup>2</sup> <sup>0</sup>

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> meets, the equilibrium <sup>E</sup><sup>1</sup> is a saddle point. Similarly, the

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>2ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> <sup>2</sup>v2ð Þ <sup>3</sup>b<sup>α</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup> <sup>∗</sup>

<sup>2</sup> <sup>v</sup>1ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup> <sup>∗</sup>

<sup>1</sup> <sup>þ</sup> ð Þ <sup>2</sup>v2<sup>B</sup> <sup>þ</sup> <sup>v</sup>1<sup>D</sup> <sup>q</sup> <sup>∗</sup>

1 � �<sup>2</sup> <sup>þ</sup> <sup>q</sup> <sup>∗</sup>

ð Þ B � D <sup>B</sup> <sup>þ</sup> <sup>D</sup> <sup>&</sup>gt;<sup>0</sup>

1 � v2ð Þ 1 � α ð Þ a � c

1

CCA,

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

<sup>3</sup>α�<sup>2</sup> and

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

1

2

1

2 � �<sup>2</sup> � � <sup>þ</sup> <sup>4</sup>v1v2B<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>v</sup>2ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup> <sup>∗</sup>

1

(13)

q ∗ <sup>1</sup> q <sup>∗</sup> 2

(14)

T evaluated at the boundary equilibrium point E<sup>1</sup> can be written as,

<sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> <sup>α</sup> � <sup>1</sup>

� <sup>v</sup>2ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> ð Þ <sup>2</sup><sup>α</sup> � <sup>1</sup> 3α � 2

The eigenvalues of J Eð Þ<sup>1</sup> are given by <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> <sup>α</sup>�<sup>1</sup>

λ<sup>2</sup> ¼ 1 � v2ð Þ 1 � α ð Þ a � c . Since (11) holds and v<sup>1</sup> >0, then λ<sup>1</sup> > 1 always meets. According to (11) and <sup>v</sup><sup>2</sup> <sup>&</sup>gt;0, we can deduce that j j <sup>λ</sup><sup>2</sup> <sup>&</sup>lt; 1 if and only if <sup>v</sup><sup>2</sup> <sup>&</sup>lt; <sup>2</sup>

Similar to the case of the equilibrium E1, it can be also proved that E<sup>2</sup> is a saddle

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , and <sup>E</sup><sup>2</sup> is an unstable node when <sup>v</sup><sup>1</sup> <sup>&</sup>gt; <sup>2</sup>

For the purpose of research of the local stability near the Nash equilibrium, we

� �

should compute the Jacobian matrix evaluated at the Nash equilibrium E<sup>∗</sup> as,

<sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup> <sup>∗</sup>

It can be seen that the form of the Jacobian matrix is so complex. In order to

A ¼ ð Þ 1 � α ð Þ a � c , B ¼ bð Þ 3α � 2 , D ¼ bð Þ 2α � 1

Then the trace and the determinant of the Jacobian matrix evaluated at the Nash

According to Jury condition, if we substitute the specific mathematical expres-

<sup>2</sup> <sup>þ</sup> <sup>2</sup>v1v2BD q <sup>∗</sup>

<sup>2</sup> into the above two equations, then the following set of inequalities

ð Þ B � D <sup>B</sup> <sup>þ</sup> <sup>D</sup> <sup>&</sup>gt;<sup>0</sup>

J Eð Þ¼ <sup>1</sup>

That is, when v<sup>2</sup> < <sup>2</sup>

point when v<sup>1</sup> < <sup>2</sup>

J q <sup>∗</sup> <sup>1</sup> ; q <sup>∗</sup> 2

sions of q <sup>∗</sup>

0≤α< <sup>3</sup> 5

45

<sup>1</sup> , q <sup>∗</sup>

0

DOI: http://dx.doi.org/10.5772/intechopen.87200

BB@

� � <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> <sup>2</sup>v1ð Þ <sup>3</sup>b<sup>α</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup> <sup>∗</sup>

simplify the calculation, let

equilibrium E<sup>∗</sup> can be given as,

equilibrium E<sup>1</sup> is an unstable node, when v<sup>2</sup> > <sup>2</sup>

<sup>v</sup>2ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup> <sup>∗</sup>

Tr <sup>¼</sup> <sup>2</sup> <sup>þ</sup> A vð Þþ <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> ð Þ <sup>2</sup>v1<sup>B</sup> <sup>þ</sup> <sup>v</sup>2<sup>D</sup> <sup>q</sup> <sup>∗</sup>

Det <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ð Þ <sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>A</sup> <sup>þ</sup> <sup>v</sup>1v2A<sup>2</sup> <sup>þ</sup> ð Þ <sup>v</sup>2<sup>D</sup> <sup>þ</sup> <sup>v</sup>1v2AD <sup>þ</sup> <sup>2</sup>v1<sup>B</sup> <sup>þ</sup> <sup>2</sup>v1v2AB <sup>q</sup> <sup>∗</sup>

<sup>4</sup> � <sup>2</sup>AB vð Þ� <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>v</sup>1v2A<sup>2</sup>

<sup>B</sup> <sup>þ</sup> <sup>D</sup> <sup>&</sup>gt; <sup>0</sup>

Since all the equilibrium points are non-negative when the parameters meet

, a>c and vi > 0, ið Þ ¼ 1; 2 . So we can get A >0, B <0, B þ D <0 and

AB vð Þ� <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>v</sup>1v2A<sup>2</sup>

<sup>v</sup>1v2A<sup>2</sup> <sup>B</sup> � <sup>D</sup>

B � D <0, then the set of inequalities (14) are equivalent to

<sup>þ</sup> ð Þ <sup>2</sup>v2<sup>B</sup> <sup>þ</sup> <sup>2</sup>v1v2AB <sup>þ</sup> <sup>v</sup>1<sup>D</sup> <sup>þ</sup> <sup>v</sup>1v2AD <sup>q</sup> <sup>∗</sup>

can be gotten through a complex calculation,

8

>>>>>>>><

>>>>>>>>:

By setting qi ð Þ¼ t þ 1 qi ð Þt , i ¼ 1, 2 in system (10), the fixed points of the system are obtained. Besides the trivial equilibrium E<sup>0</sup> ¼ ð Þ 0; 0 , system (10) admits the following non-trivial fixed points (boundary equilibrium points),

$$E\_1 = \left( \mathbf{0}, -\frac{(a-c)A\_2}{bB\_2} \right), \ E\_2 = \left( -\frac{(a-c)A\_1}{bB\_1}, \mathbf{0} \right).$$

and the only Nash equilibrium is

$$E^\* = \left(\frac{(a-c)(A\_2\mathbf{C}\_1 - A\_1\mathbf{B}\_2)}{b(B\_1\mathbf{B}\_2 - \mathbf{C}\_1\mathbf{C}\_2)}, \frac{(a-c)(A\_1\mathbf{C}\_2 - A\_2\mathbf{B}\_1)}{b(B\_1\mathbf{B}\_2 - \mathbf{C}\_1\mathbf{C}\_2)}\right)^2$$

where A<sup>1</sup> ¼ 1 � α1, A<sup>2</sup> ¼ 1 � α2, B<sup>1</sup> ¼ 3α<sup>1</sup> � 2, B<sup>2</sup> ¼ 3α<sup>2</sup> � 2, C<sup>1</sup> ¼ 2α<sup>1</sup> � 1 and <sup>C</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>α<sup>2</sup> � 1. The positivity of Eið Þ <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup> and <sup>E</sup><sup>∗</sup> is ensured by requiring S ¼ S<sup>1</sup> ∪ S2, where

$$\begin{cases} \mathcal{S}\_1 = \left\{ (A\_1, A\_1, B\_1, B\_2, C\_1, C\_2) | B\_1 < 0, B\_2 < 0, B\_1 B\_2 - C\_1 C\_2 > 0, A\_2 C\_1 - A\_1 B\_2 > 0, A\_1 C\_2 - A\_2 B\_1 > 0 \right\} \\ \mathcal{S}\_2 = \left\{ (A\_1, A\_1, B\_1, B\_2, C\_1, C\_2) | B\_1 < 0, B\_2 < 0, B\_1 B\_2 - C\_1 C\_2 < 0, A\_2 C\_1 - A\_1 B\_2 < 0, A\_1 C\_2 - A\_2 B\_1 < 0 \right\} \end{cases} \tag{11}$$

#### 3. Stability properties

The local stability analyses of system (10) near the fixed points are too difficult to carry on. For the sake of analyzing the local stability of the system, we firstly let α<sup>1</sup> ¼ α<sup>2</sup> ¼ α in system (10). And the Jacobian matrix of map T at any fixed point q1; q<sup>2</sup> can be given as,

$$f(q\_1, q\_2) = \begin{pmatrix} 1 + \nu\_1(1 - a)(a - \varepsilon) + 2\nu\_1(3ba - 2b)q\_1 + \nu\_1(2ba - b)q\_2 & \nu\_1(2ba - b)q\_1 \\ \nu\_2(2ba - b)q\_2 & 1 + \nu\_2(1 - a)(a - \varepsilon) + 2\nu\_2(3ba - 2b)q\_2 + \nu\_2(2ba - b)q\_1 \end{pmatrix} \tag{12}$$

Then all the equilibrium points are substituted into the Jacobian matrix (12). According to the characteristic values of the Jacobian matrix evaluated at each equilibrium, the type and stability of the equilibrium can be analyzed and the following results can be obtained.

Proposition 1. The equilibrium point E<sup>0</sup> is always an unstable node.

Proof. It is clear that the Jacobian matrix of map T, evaluated at the boundary equilibrium point E<sup>0</sup> can be written as,

$$J(E\_0) = \begin{pmatrix} \mathbf{1} + \nu\_1(\mathbf{1} - a)(a - c) & \mathbf{0} \\ \mathbf{0} & \mathbf{1} + \nu\_2(\mathbf{1} - a)(a - c) \end{pmatrix},$$

The eigenvalues of J Eð Þ<sup>0</sup> are given by λ<sup>i</sup> ¼ 1 þ við Þ 1 � α ð Þ a � c , i ¼ 1, 2. Since (11) holds and vi >0, ið Þ ¼ 1; 2 then λ<sup>i</sup> > 1, ið Þ ¼ 1; 2 which implies that E<sup>0</sup> is an unstable node.

Proposition 2. E<sup>1</sup> is a saddle point, when v<sup>2</sup> < <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> . And <sup>E</sup><sup>1</sup> is an unstable node, when v<sup>2</sup> > <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

Proof. By substituting the equilibrium E<sup>1</sup> into (11), the Jacobian matrix of map T evaluated at the boundary equilibrium point E<sup>1</sup> can be written as,

$$J(E\_1) = \begin{pmatrix} \lambda\_1 = \mathbf{1} + \nu\_1(\mathbf{1} - a)(a - c) \frac{a - \mathbf{1}}{3a - \mathbf{2}} & \mathbf{0} \\ -\frac{\nu\_2(\mathbf{1} - a)(a - c)(2a - \mathbf{1})}{3a - \mathbf{2}} & \mathbf{1} - \nu\_2(\mathbf{1} - a)(a - c) \end{pmatrix},$$

The eigenvalues of J Eð Þ<sup>1</sup> are given by <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> <sup>α</sup>�<sup>1</sup> <sup>3</sup>α�<sup>2</sup> and λ<sup>2</sup> ¼ 1 � v2ð Þ 1 � α ð Þ a � c . Since (11) holds and v<sup>1</sup> >0, then λ<sup>1</sup> > 1 always meets. According to (11) and <sup>v</sup><sup>2</sup> <sup>&</sup>gt;0, we can deduce that j j <sup>λ</sup><sup>2</sup> <sup>&</sup>lt; 1 if and only if <sup>v</sup><sup>2</sup> <sup>&</sup>lt; <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> . That is, when v<sup>2</sup> < <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> meets, the equilibrium <sup>E</sup><sup>1</sup> is a saddle point. Similarly, the equilibrium E<sup>1</sup> is an unstable node, when v<sup>2</sup> > <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

Similar to the case of the equilibrium E1, it can be also proved that E<sup>2</sup> is a saddle point when v<sup>1</sup> < <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , and <sup>E</sup><sup>2</sup> is an unstable node when <sup>v</sup><sup>1</sup> <sup>&</sup>gt; <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

For the purpose of research of the local stability near the Nash equilibrium, we should compute the Jacobian matrix evaluated at the Nash equilibrium E<sup>∗</sup> as,

$$J(q\_1^\*, q\_2^\*) = \begin{pmatrix} 1 + \nu\_1(1 - a)(a - \varepsilon) + 2\nu\_1(3ba - 2b)q\_1^\* + \nu\_1(2ba - b)q\_2^\* & \nu\_1(2ba - b)q\_1^\* \\ \nu\_2(2ba - b)q\_2^\* & 1 + \nu\_2(1 - a)(a - \varepsilon) + 2\nu\_2(3ba - 2b)q\_2^\* + \nu\_2(2ba - b)q\_1^\* \end{pmatrix} \tag{13}$$

It can be seen that the form of the Jacobian matrix is so complex. In order to simplify the calculation, let

$$A = (\mathbf{1} - a)(a - c), B = b(\mathbf{3}a - \mathbf{2}), D = b(\mathbf{2}a - \mathbf{1})$$

Then the trace and the determinant of the Jacobian matrix evaluated at the Nash equilibrium E<sup>∗</sup> can be given as,

$$Tr = 2 + A(\nu\_1 + \nu\_2) + (2\nu\_1 B + \nu\_2 D)q\_1^\* + (2\nu\_2 B + \nu\_1 D)q\_2^\*$$

$$\begin{split} \text{Det} &= 1 + (\nu\_1 + \nu\_2)A + \nu\_1 \nu\_2 A^2 + (\nu\_2 D + \nu\_1 \nu\_2 AD + 2\nu\_1 B + 2\nu\_1 \nu\_2 AB)q\_1^\* \\ &+ (2\nu\_2 B + 2\nu\_1 \nu\_2 AB + \nu\_1 D + \nu\_1 \nu\_2 AD)q\_2^\* + 2\nu\_1 \nu\_2 BD \left( \left(q\_1^\*\right)^2 + \left(q\_2^\*\right)^2 \right) + 4\nu\_1 \nu\_2 B^2 q\_1^\* q\_2^\* \end{split}$$

According to Jury condition, if we substitute the specific mathematical expressions of q <sup>∗</sup> <sup>1</sup> , q <sup>∗</sup> <sup>2</sup> into the above two equations, then the following set of inequalities can be gotten through a complex calculation,

$$\begin{cases} 4 - \frac{2AB(v\_1 + v\_2) - v\_1v\_2A^2(B - D)}{B + D} > 0 \\\\ v\_1v\_2A^2 \frac{B - D}{B + D} > 0 \\\\ \frac{AB(v\_1 + v\_2) - v\_1v\_2A^2(B - D)}{B + D} > 0 \end{cases} \tag{14}$$

Since all the equilibrium points are non-negative when the parameters meet 0≤α< <sup>3</sup> 5 , a>c and vi > 0, ið Þ ¼ 1; 2 . So we can get A >0, B <0, B þ D <0 and B � D <0, then the set of inequalities (14) are equivalent to

F ¼ q1; q<sup>2</sup>

<sup>E</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>; � ð Þ <sup>a</sup> � <sup>c</sup> <sup>A</sup><sup>2</sup>

<sup>E</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>a</sup> � <sup>c</sup> ð Þ <sup>A</sup>2C<sup>1</sup> � <sup>A</sup>1B<sup>2</sup> b Bð Þ <sup>1</sup>B<sup>2</sup> � C1C<sup>2</sup>

ð Þ¼ t þ 1 qi

and the only Nash equilibrium is

By setting qi

Research Advances in Chaos Theory

S ¼ S<sup>1</sup> ∪ S2, where

3. Stability properties

can be given as,

following results can be obtained.

equilibrium point E<sup>0</sup> can be written as,

J Eð Þ¼ <sup>0</sup>

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> .

Proposition 2. E<sup>1</sup> is a saddle point, when v<sup>2</sup> < <sup>2</sup>

unstable node.

44

node, when v<sup>2</sup> > <sup>2</sup>

q1; q<sup>2</sup>

J q1; q<sup>2</sup>

: <sup>q</sup><sup>1</sup> <sup>≥</sup> <sup>0</sup>; <sup>q</sup><sup>2</sup> <sup>≥</sup>0; <sup>q</sup><sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> 6¼ <sup>0</sup>

;

where A<sup>1</sup> ¼ 1 � α1, A<sup>2</sup> ¼ 1 � α2, B<sup>1</sup> ¼ 3α<sup>1</sup> � 2, B<sup>2</sup> ¼ 3α<sup>2</sup> � 2, C<sup>1</sup> ¼ 2α<sup>1</sup> � 1 and

S<sup>1</sup> ¼ fð Þ A1; A1; B1; B2;C1;C<sup>2</sup> j g B<sup>1</sup> <0; B<sup>2</sup> <0; B1B<sup>2</sup> � C1C<sup>2</sup> >0; A2C<sup>1</sup> � A1B<sup>2</sup> >0; A1C<sup>2</sup> � A2B<sup>1</sup> >0 S<sup>2</sup> ¼ fð Þ A1; A1; B1; B2;C1;C<sup>2</sup> j g B<sup>1</sup> <0; B<sup>2</sup> <0; B1B<sup>2</sup> � C1C<sup>2</sup> <0; A2C<sup>1</sup> � A1B<sup>2</sup> <0; A1C<sup>2</sup> � A2B<sup>1</sup> <0

The local stability analyses of system (10) near the fixed points are too difficult to carry on. For the sake of analyzing the local stability of the system, we firstly let α<sup>1</sup> ¼ α<sup>2</sup> ¼ α in system (10). And the Jacobian matrix of map T at any fixed point

Then all the equilibrium points are substituted into the Jacobian matrix (12). According to the characteristic values of the Jacobian matrix evaluated at each equilibrium, the type and stability of the equilibrium can be analyzed and the

Proof. It is clear that the Jacobian matrix of map T, evaluated at the boundary

1 þ v1ð Þ 1 � α ð Þ a � c 0

The eigenvalues of J Eð Þ<sup>0</sup> are given by λ<sup>i</sup> ¼ 1 þ við Þ 1 � α ð Þ a � c , i ¼ 1, 2. Since (11) holds and vi >0, ið Þ ¼ 1; 2 then λ<sup>i</sup> > 1, ið Þ ¼ 1; 2 which implies that E<sup>0</sup> is an

0 1 þ v2ð Þ 1 � α ð Þ a � c

v2ð Þ 2bα � b q<sup>2</sup> 1 þ v2ð Þ 1 � α ð Þþ a � c 2v2ð Þ 3bα � 2b q<sup>2</sup> þ v2ð Þ 2bα � b q<sup>1</sup>

<sup>C</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>α<sup>2</sup> � 1. The positivity of Eið Þ <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup> and <sup>E</sup><sup>∗</sup> is ensured by requiring

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> <sup>2</sup>v1ð Þ <sup>3</sup>b<sup>α</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup><sup>1</sup> <sup>þ</sup> <sup>v</sup>1ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup><sup>2</sup> <sup>v</sup>1ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup><sup>1</sup>

Proposition 1. The equilibrium point E<sup>0</sup> is always an unstable node.

are obtained. Besides the trivial equilibrium E<sup>0</sup> ¼ ð Þ 0; 0 , system (10) admits the

following non-trivial fixed points (boundary equilibrium points),

bB<sup>2</sup> 

ð Þt , i ¼ 1, 2 in system (10), the fixed points of the system

bB<sup>1</sup>

ð Þ a � c ð Þ A1C<sup>2</sup> � A2B<sup>1</sup> b Bð Þ <sup>1</sup>B<sup>2</sup> � C1C<sup>2</sup>

; 0

(11)

(12)

,

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> . And <sup>E</sup><sup>1</sup> is an unstable

, E<sup>2</sup> ¼ � ð Þ <sup>a</sup> � <sup>c</sup> <sup>A</sup><sup>1</sup>

$$\begin{cases} 4(B+D) - 2AB(v\_1+v\_2) + v\_1v\_2A^2(B-D) < 0\\ B(v\_1+v\_2) - v\_1v\_2A(B-D) < 0 \end{cases} \tag{15}$$

black region. It means that when the firms change their speed of adjustment according to the path, a periodic fluctuation of system (10) will happen. That is, the period motion will increase exponentially until it enters chaos. And the chaotic behavior of this system can be understood as the confusion of the market competition, and one of the two firms may be even out of the market with an increasing speed of adjustment. But if the parameter passes through the green region from the brown region to the black region, the system will first undergo a flip bifurcation, and then enters quasi-period motion through a Neimark-Sacker bifurcation. The system enters quasi-period from period-2 when firms determine their speed of adjustment along this path. Figure 1b is a partial amplification of Figure 1a, and we can observe that there are many scattered points of different colors, which is caused

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

Figure 2a shows the coexistence of attractors with the parameters chosen as v<sup>1</sup> ¼ 2:44, and v<sup>2</sup> ¼ 2:45, where the scatter points are shown in Figure 1b. we can observe a period-6 cycle coexisting with a period-4 cycle. Figure 1c is the twodimensional bifurcation diagram, where the fixed parameters are given as

a � c ¼ 0:88017028, α ¼ 0:27445462 and b ¼ 0:52714274. At this set of parameters, the system enters chaos through a flip bifurcation. Figure 1d is a partial enlargement of Figure 1c. Similarly, the parameter space of Figure 1d is also chosen according to the area with scattered points of Figure 1c. Figure 2b shows the coexisted attractors and their basins of attraction at this set of parameters. We can

Figure 3 shows a series of two-dimensional bifurcation diagram under different parameters. It shows a very beautiful gallery, from which we can enjoy the system (10) with full complex dynamics phenomenon. We can observe from Figure 3 that the difference between the maximum price of the unit product a and the marginal cost c affects the size of the stable region. The weight of the consumer surplus α affects the shape of the two-dimensional bifurcation diagram, while the parameter b almost has hardly effect on the two-dimensional bifurcation of the system.

Therefore, the game can be balanced more quickly by reducing the difference a � c. In Figure 3a, it can be observed that the chaotic area surrounded by the period-4 area is like a "hand" and the 8-period area is like a small "bottle" raised by the beautiful hand. It is observed that the shape of Figure 3b is similar to Figure 3a due to a tiny adjustment of parameter α. Since the difference between the parameters a

and c is reduced, the stability area of the Nash equilibrium becomes larger. Figure 3c is like a "volcanic eruption." It can be observed that there is an inward

(a) A period-8 cycle coexists with a period-8 cycle with a � c ¼ 2.1, α ¼ 0.50608821, b ¼ 2.00232504, v1 ¼ 2.44, and v2 ¼ 2.45 and (b) a attractor on the diagonal and a period-4 cycle coexists with a period-8 cycle with a � c ¼ 0.88017028, α ¼ 0.27445462, v1 ¼ 4.16, b ¼ 0.52714274, and v2 ¼ 4.13.

by the coexistence of multi-attractors with different period.

DOI: http://dx.doi.org/10.5772/intechopen.87200

observe that there are three attractors coexisting.

Figure 2.

47

Then the stability region of the Nash equilibrium can be obtained by substituting A, B and D into inequalities (15), which are given as,

$$\begin{cases} 4(5a-3) - 2(1-a)(a-c)(3a-2)(v\_1+v\_2) - v\_1v\_2(1-a)^3(a-c)^2 < 0\\ (3a-2)(v\_1+v\_2) + v\_1v\_2(1-a)^2(a-c) < 0 \end{cases} \tag{16}$$

The stability condition of the Nash equilibrium gives a parameters region, in which the Nash equilibrium is always stable. For the sake of better analysis of the stability of the Nash equilibrium under different set of parameters, a useful tool called "two-dimensional bifurcation diagram" (also called 2-D bifurcation diagram) is employed. From (16), we can find that the stability region of the Nash equilibrium is related to the difference of parameters a and c, that is a � c. So we only discuss the values of a � c, rather than the values of a and c in the rest of this chapter.

Figure 1 is a two-dimensional bifurcation diagram of system (10) with a set of fixed parameter a � c, b and α. Figure 1a is a two-dimensional bifurcation diagram when a � c ¼ 2:1, α ¼ 0:50608821 and b ¼ 2:00232504. We can observe that there are two different routes to chaos, when the parameters are chosen as this set of parameters. The system enters chaos through flip bifurcation when the parameter ð Þ v1; v<sup>2</sup> passes through green, yellow and light green from the brown region to the

#### Figure 1.

2D bifurcation diagram in the v1 ð Þ ; v2 or α<sup>1</sup> ð Þ ; α<sup>2</sup> parameters plane. (a) a � c ¼ 2.1, α ¼ 0.50608821, b ¼ 2.00232504, (b) enlarged the square region in Figure 1a, (c) a � c ¼ 0.88017028, α ¼ 0.27445462, b ¼ 0.52714274 and (d) enlarged the square region in the Figure 1c.

#### Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

black region. It means that when the firms change their speed of adjustment according to the path, a periodic fluctuation of system (10) will happen. That is, the period motion will increase exponentially until it enters chaos. And the chaotic behavior of this system can be understood as the confusion of the market competition, and one of the two firms may be even out of the market with an increasing speed of adjustment. But if the parameter passes through the green region from the brown region to the black region, the system will first undergo a flip bifurcation, and then enters quasi-period motion through a Neimark-Sacker bifurcation. The system enters quasi-period from period-2 when firms determine their speed of adjustment along this path. Figure 1b is a partial amplification of Figure 1a, and we can observe that there are many scattered points of different colors, which is caused by the coexistence of multi-attractors with different period.

Figure 2a shows the coexistence of attractors with the parameters chosen as v<sup>1</sup> ¼ 2:44, and v<sup>2</sup> ¼ 2:45, where the scatter points are shown in Figure 1b. we can observe a period-6 cycle coexisting with a period-4 cycle. Figure 1c is the twodimensional bifurcation diagram, where the fixed parameters are given as a � c ¼ 0:88017028, α ¼ 0:27445462 and b ¼ 0:52714274. At this set of parameters, the system enters chaos through a flip bifurcation. Figure 1d is a partial enlargement of Figure 1c. Similarly, the parameter space of Figure 1d is also chosen according to the area with scattered points of Figure 1c. Figure 2b shows the coexisted attractors and their basins of attraction at this set of parameters. We can observe that there are three attractors coexisting.

Figure 3 shows a series of two-dimensional bifurcation diagram under different parameters. It shows a very beautiful gallery, from which we can enjoy the system (10) with full complex dynamics phenomenon. We can observe from Figure 3 that the difference between the maximum price of the unit product a and the marginal cost c affects the size of the stable region. The weight of the consumer surplus α affects the shape of the two-dimensional bifurcation diagram, while the parameter b almost has hardly effect on the two-dimensional bifurcation of the system. Therefore, the game can be balanced more quickly by reducing the difference a � c. In Figure 3a, it can be observed that the chaotic area surrounded by the period-4 area is like a "hand" and the 8-period area is like a small "bottle" raised by the beautiful hand. It is observed that the shape of Figure 3b is similar to Figure 3a due to a tiny adjustment of parameter α. Since the difference between the parameters a and c is reduced, the stability area of the Nash equilibrium becomes larger. Figure 3c is like a "volcanic eruption." It can be observed that there is an inward

#### Figure 2.

(a) A period-8 cycle coexists with a period-8 cycle with a � c ¼ 2.1, α ¼ 0.50608821, b ¼ 2.00232504, v1 ¼ 2.44, and v2 ¼ 2.45 and (b) a attractor on the diagonal and a period-4 cycle coexists with a period-8 cycle with a � c ¼ 0.88017028, α ¼ 0.27445462, v1 ¼ 4.16, b ¼ 0.52714274, and v2 ¼ 4.13.

<sup>4</sup>ð Þ� <sup>B</sup> <sup>þ</sup> <sup>D</sup> <sup>2</sup>AB vð Þþ <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>v</sup>1v2A<sup>2</sup>

Then the stability region of the Nash equilibrium can be obtained by substituting

ð Þ a � c <0

The stability condition of the Nash equilibrium gives a parameters region, in which the Nash equilibrium is always stable. For the sake of better analysis of the stability of the Nash equilibrium under different set of parameters, a useful tool called "two-dimensional bifurcation diagram" (also called 2-D bifurcation diagram) is employed. From (16), we can find that the stability region of the Nash equilibrium is related to the difference of parameters a and c, that is a � c. So we only discuss the

Figure 1 is a two-dimensional bifurcation diagram of system (10) with a set of fixed parameter a � c, b and α. Figure 1a is a two-dimensional bifurcation diagram when a � c ¼ 2:1, α ¼ 0:50608821 and b ¼ 2:00232504. We can observe that there are two different routes to chaos, when the parameters are chosen as this set of parameters. The system enters chaos through flip bifurcation when the parameter ð Þ v1; v<sup>2</sup> passes through green, yellow and light green from the brown region to the

values of a � c, rather than the values of a and c in the rest of this chapter.

2D bifurcation diagram in the v1 ð Þ ; v2 or α<sup>1</sup> ð Þ ; α<sup>2</sup> parameters plane. (a) a � c ¼ 2.1, α ¼ 0.50608821, b ¼ 2.00232504, (b) enlarged the square region in Figure 1a, (c) a � c ¼ 0.88017028, α ¼ 0.27445462,

b ¼ 0.52714274 and (d) enlarged the square region in the Figure 1c.

B vð Þ� <sup>1</sup> þ v<sup>2</sup> v1v2A Bð Þ � D <0

4 5ð Þ� <sup>α</sup> � <sup>3</sup> 2 1ð Þ � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> ð Þ <sup>3</sup><sup>α</sup> � <sup>2</sup> ð Þ� <sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>v</sup>1v2ð Þ <sup>1</sup> � <sup>α</sup> <sup>3</sup>

A, B and D into inequalities (15), which are given as,

(

Research Advances in Chaos Theory

ð Þ <sup>3</sup><sup>α</sup> � <sup>2</sup> ð Þþ <sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>v</sup>1v2ð Þ <sup>1</sup> � <sup>α</sup> <sup>2</sup>

(

Figure 1.

46

ð Þ B � D < 0

ð Þ a � c

<sup>2</sup> <0

(15)

(16)

Subsequently, we assume that both firms have the same speed of adjustment. It means that the latter discussion is based on v<sup>1</sup> ¼ v<sup>2</sup> ¼ v and α<sup>1</sup> ¼ α<sup>2</sup> ¼ α. In this case the two players are identical; the system T can then be rewritten as follows,

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

<sup>T</sup><sup>0</sup> : <sup>q</sup>1ð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>1</sup> <sup>q</sup>1ðÞþ<sup>t</sup> vq1ð Þ<sup>t</sup> ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> ð Þ <sup>3</sup>b<sup>α</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup>1ðÞþ<sup>t</sup> ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup>2ð Þ<sup>t</sup> � � <sup>q</sup>2ð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>1</sup> <sup>q</sup>2ðÞþ<sup>t</sup> vq2ð Þ<sup>t</sup> ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> ð Þ <sup>3</sup>b<sup>α</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup>2ðÞþ<sup>t</sup> ð Þ <sup>2</sup>b<sup>α</sup> � <sup>b</sup> <sup>q</sup>1ð Þ<sup>t</sup> � � (

It can be proved that the map T<sup>0</sup> has symmetry property, i.e., there exists a map

∘S ¼ S∘T<sup>0</sup>

ð Þ Δ ⊆Δ. However, the phenomenon of synchronization occurs when the diagonal Δ is invariant one-dimensional submanifold of the system (17). Therefore, the phenomenon of synchronization of the system can be analyzed by studying the invariant set. We can also use critical curve and noninvertible map to describe the

T<sup>0</sup> implies that the diagonal Δ is a one-dimensional sub-manifold of system (17), i.e.,

We divide the discrete dynamical system into invertible and noninvertible. The

implies that the map T<sup>0</sup> is multi-valued, i.e., the image of T<sup>0</sup> has more than one preimages. In a noninvertible discrete dynamical system, the curve that divides the phase space into regions with a different number of rank-1 preimages is called critical curve, denoted by LC. And the regions can be represented by Zi, ið Þ ∈ N . For example, a point belonging to area Z<sup>0</sup> has no preimage and a point belonging to area Z<sup>2</sup> has two preimages. Let us denote the rank-1 preimages of critical curve LC under map T<sup>0</sup> as LC�1. The set LC is the 2-dimensional generalization of the critical value or local extremes of 1-dimensional noninvertible map. Its preimages LC�<sup>1</sup> are corresponding to local extreme point (critical point) in the one-dimensional noninvertible map. Since the map (17) is a continuously differentiable map, LC�<sup>1</sup> belongs to the locus of points

S : q1; q<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>2</sup>vA <sup>þ</sup> <sup>v</sup><sup>2</sup>

shown in Figure 10f.

4.2 Invariant sets

rewritten as,

49

T0

� � ! <sup>q</sup>2; <sup>q</sup><sup>1</sup>

� �, which makes T<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.87200

global dynamical behaviors of a 2-dimensional map.

invertible discrete dynamical system refers that an image q<sup>1</sup>

where the Jacobian determinant ofT<sup>0</sup> vanishes, i.e., LC�<sup>1</sup> ¼ q1; q<sup>2</sup>

AD <sup>þ</sup> <sup>2</sup>vB <sup>þ</sup> <sup>2</sup>v<sup>2</sup> AB � � <sup>q</sup><sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

LC is the rank-1 image of LC�<sup>1</sup> under map T<sup>0</sup>

In this case, curve LC�<sup>1</sup> can be determined by the following equation,

4.1 Critical curve and noninvertible map

correspond to the only preimage q1; q<sup>2</sup>

<sup>A</sup><sup>2</sup> <sup>þ</sup> vD <sup>þ</sup> <sup>v</sup><sup>2</sup>

(17)

. The symmetry property of the map

0 ; q<sup>2</sup>

BD q<sup>2</sup>

. That is, LC ¼ T<sup>0</sup>

<sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> 2 � � <sup>þ</sup> <sup>4</sup>v<sup>2</sup>

� �. The noninvertible discrete dynamical system

� � <sup>þ</sup> <sup>2</sup>v<sup>2</sup>

noninvertible properties play a significant role in analyzing the global behavior of a nonlinear discrete dynamical model. So the critical curve is a powerful tool for us to study these complex structures. Using the segment of critical curve as well as their preimages of any rank, and we will get the boundary of the basins of attraction as

The dynamics of the system on the diagonal is studied by analyzing the invariant

sets. Firstly, we can prove that the coordinates are invariant sets of map T<sup>0</sup>

q2ðÞ¼ t 0, then we can obtain q2ð Þ¼ t þ 1 0, and the first equation of (17) can be

<sup>0</sup> � � of the map T<sup>0</sup> is

� �∈R<sup>2</sup> detDT<sup>0</sup> j g <sup>¼</sup> <sup>0</sup> � .

ð Þ LC�<sup>1</sup> . The

Bq1q<sup>2</sup> ¼ 0 (18)

. Let

Figure 3.

2D bifurcation diagram in the (v1, v2) parameters plane, (a) with parameters a � c ¼ 1.83, α ¼ 0.278, b ¼ 0.0012, (b) with parameters a � c ¼ 1.52, α ¼ 0.277, b ¼ 0.348, (c) with parameters a � c ¼ 1.66, α ¼ 0.27, b ¼ 0.19 and (d) with parameters a � c ¼ 1.99, α ¼ 0.39, b ¼ 0.22.

cave in the diagonal that is like a "crater." As the parameters vary, the hole in Figure 3c continues to sink inward and become larger and larger. From Figure 3d we can see that the period-4 arrives at a quasi-periodic motion directly. Therefore, we should change the weight of consumer surplus α slightly in order to maintain the market fluctuations not fierceness.

#### 4. Global dynamics and synchronization

The type and stability of the equilibrium points have been analyzed as above. And the boundary equilibrium E<sup>1</sup> and the boundary equilibrium E<sup>2</sup> are in symmetrical positions with respect to the main diagonal line Δ ¼ q1; q<sup>2</sup> ∈R<sup>2</sup> <sup>þ</sup> : q<sup>1</sup> ¼ q<sup>2</sup> . It is also clear that the unique Nash equilibrium E<sup>∗</sup> of system (10) is located on the main diagonal Δ. So we mainly study the dynamical behavior of the system on the diagonal. We choose the initial conditions near the diagonal, and the phenomenon via finite iteration back to the diagonal is called synchronization. The synchronization of chaotic systems was quite interesting and unexpected. In fact, due to the nonlinear system usually has sensitive dependence on initial conditions, a property which implies that the slightly change of initial conditions will lead to an exponential difference between the trajectories of two identical systems, making it impossible for two separated and even identical systems to synchronize. Therefore, the small coupling between two chaotic oscillators makes the system asymptotically converge to same trajectory, which is worth studying.

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

Subsequently, we assume that both firms have the same speed of adjustment. It means that the latter discussion is based on v<sup>1</sup> ¼ v<sup>2</sup> ¼ v and α<sup>1</sup> ¼ α<sup>2</sup> ¼ α. In this case the two players are identical; the system T can then be rewritten as follows,

$$T': \begin{cases} q\_1(t+1) = q\_1(t) + \nu q\_1(t) \left[ (1-a)(a-c) + (3ba-2b)q\_1(t) + (2ba-b)q\_2(t) \right] \\ q\_2(t+1) = q\_2(t) + \nu q\_2(t) \left[ (1-a)(a-c) + (3ba-2b)q\_2(t) + (2ba-b)q\_1(t) \right] \end{cases} \tag{17}$$

It can be proved that the map T<sup>0</sup> has symmetry property, i.e., there exists a map S : q1; q<sup>2</sup> � � ! <sup>q</sup>2; <sup>q</sup><sup>1</sup> � �, which makes T<sup>0</sup> ∘S ¼ S∘T<sup>0</sup> . The symmetry property of the map T<sup>0</sup> implies that the diagonal Δ is a one-dimensional sub-manifold of system (17), i.e., T0 ð Þ Δ ⊆Δ. However, the phenomenon of synchronization occurs when the diagonal Δ is invariant one-dimensional submanifold of the system (17). Therefore, the phenomenon of synchronization of the system can be analyzed by studying the invariant set. We can also use critical curve and noninvertible map to describe the global dynamical behaviors of a 2-dimensional map.

#### 4.1 Critical curve and noninvertible map

We divide the discrete dynamical system into invertible and noninvertible. The invertible discrete dynamical system refers that an image q<sup>1</sup> 0 ; q<sup>2</sup> <sup>0</sup> � � of the map T<sup>0</sup> is correspond to the only preimage q1; q<sup>2</sup> � �. The noninvertible discrete dynamical system implies that the map T<sup>0</sup> is multi-valued, i.e., the image of T<sup>0</sup> has more than one preimages. In a noninvertible discrete dynamical system, the curve that divides the phase space into regions with a different number of rank-1 preimages is called critical curve, denoted by LC. And the regions can be represented by Zi, ið Þ ∈ N . For example, a point belonging to area Z<sup>0</sup> has no preimage and a point belonging to area Z<sup>2</sup> has two preimages. Let us denote the rank-1 preimages of critical curve LC under map T<sup>0</sup> as LC�1. The set LC is the 2-dimensional generalization of the critical value or local extremes of 1-dimensional noninvertible map. Its preimages LC�<sup>1</sup> are corresponding to local extreme point (critical point) in the one-dimensional noninvertible map. Since the map (17) is a continuously differentiable map, LC�<sup>1</sup> belongs to the locus of points where the Jacobian determinant ofT<sup>0</sup> vanishes, i.e., LC�<sup>1</sup> ¼ q1; q<sup>2</sup> � �∈R<sup>2</sup> detDT<sup>0</sup> j g <sup>¼</sup> <sup>0</sup> � . In this case, curve LC�<sup>1</sup> can be determined by the following equation,

$$\left(\mathbf{1} + 2\nu\mathbf{A} + \nu^2\mathbf{A}^2 + \left(\nu\mathbf{D} + \nu^2\mathbf{A}\mathbf{D} + 2\nu\mathbf{B} + 2\nu^2\mathbf{A}\mathbf{B}\right)\left(q\_1 + q\_2\right) + 2\nu^2\mathbf{B}\mathbf{D}\left(q\_1^2 + q\_2^2\right) + 4\nu^2\mathbf{B}q\_1q\_2 = \mathbf{0} \tag{18}$$

LC is the rank-1 image of LC�<sup>1</sup> under map T<sup>0</sup> . That is, LC ¼ T<sup>0</sup> ð Þ LC�<sup>1</sup> . The noninvertible properties play a significant role in analyzing the global behavior of a nonlinear discrete dynamical model. So the critical curve is a powerful tool for us to study these complex structures. Using the segment of critical curve as well as their preimages of any rank, and we will get the boundary of the basins of attraction as shown in Figure 10f.

#### 4.2 Invariant sets

The dynamics of the system on the diagonal is studied by analyzing the invariant sets. Firstly, we can prove that the coordinates are invariant sets of map T<sup>0</sup> . Let q2ðÞ¼ t 0, then we can obtain q2ð Þ¼ t þ 1 0, and the first equation of (17) can be rewritten as,

cave in the diagonal that is like a "crater." As the parameters vary, the hole in Figure 3c continues to sink inward and become larger and larger. From Figure 3d we can see that the period-4 arrives at a quasi-periodic motion directly. Therefore, we should change the weight of consumer surplus α slightly in order to maintain the

2D bifurcation diagram in the (v1, v2) parameters plane, (a) with parameters a � c ¼ 1.83, α ¼ 0.278, b ¼ 0.0012, (b) with parameters a � c ¼ 1.52, α ¼ 0.277, b ¼ 0.348, (c) with parameters a � c ¼ 1.66,

The type and stability of the equilibrium points have been analyzed as above. And the boundary equilibrium E<sup>1</sup> and the boundary equilibrium E<sup>2</sup> are in symmet-

is also clear that the unique Nash equilibrium E<sup>∗</sup> of system (10) is located on the main diagonal Δ. So we mainly study the dynamical behavior of the system on the diagonal. We choose the initial conditions near the diagonal, and the phenomenon via finite iteration back to the diagonal is called synchronization. The synchronization of chaotic systems was quite interesting and unexpected. In fact, due to the nonlinear system usually has sensitive dependence on initial conditions, a property which implies that the slightly change of initial conditions will lead to an exponential difference between the trajectories of two identical systems, making it impossible for two separated and even identical systems to synchronize. Therefore, the small coupling between two chaotic oscillators makes the system asymptotically

∈R<sup>2</sup>

<sup>þ</sup> : q<sup>1</sup> ¼ q<sup>2</sup> . It

market fluctuations not fierceness.

Research Advances in Chaos Theory

Figure 3.

48

4. Global dynamics and synchronization

rical positions with respect to the main diagonal line Δ ¼ q1; q<sup>2</sup>

α ¼ 0.27, b ¼ 0.19 and (d) with parameters a � c ¼ 1.99, α ¼ 0.39, b ¼ 0.22.

converge to same trajectory, which is worth studying.

$$q\_1(t+1) = q\_1(t) + \nu q\_1(t)^{\frac{1}{2}[(1-a)(a-c) + (3ba-2b)q\_1(t)]} \tag{19}$$

It is easy to verify that the dynamics on the axis q<sup>2</sup> is also controlled by the map (19). It means that the system (17) can be regarded as a 1-dimensional map at the coordinate axes. The map (19) is topologically conjugate to the standard logistic map x tð Þ¼ þ 1 ωx tð Þð Þ 1 � x tð Þ through a linear transformation, which is given as,

$$q\_1 = \left[\frac{\mathbf{1} + v(\mathbf{1} - a)(a - c)}{v(2b - 3ba)}\right] \mathbf{x} \tag{20}$$

and the parameter ω can be presented as ω ¼ 1 þ vð Þ 1 � α ð Þ a � c . Thus the nonlinear dynamics of system (17) on the invariant axes can be analyzed through the standard logistic map.

It can also be proved that the diagonal Δ is an invariant set of system (17), i.e., the trajectory starting from the diagonal Δ will stay forever on it. Therefore, the dynamical behavior of system (17) can be analyzed through the map T<sup>0</sup> which is restricted to the diagonal. If we let q<sup>1</sup> ¼ q<sup>2</sup> ¼ q, then the dynamics generated by T<sup>0</sup> on the diagonal Δ can be analyzed through the following map,

$$T\_{\Delta}^{\prime} : q(t+1) = q(t) + \nu q(t)((1-a)(a-c) + (\mathsf{S}ba - \mathsf{3}b)q(t))\tag{21}$$

Similarly, through the following linear transformation

$$q = \left[\frac{\mathbf{1} + v(\mathbf{1} - a)(a - c)}{v(\mathbf{3}b - \mathbf{5}ba)}\right] \mathbf{y} \tag{22}$$

upper area lead to the divergence of the trajectory, corresponding to the black area in Figure 4b. Through the above analysis, the following proposition can be

and (b) 2D bifurcation diagram in the parameter plane (α, v) for a � c ¼ 2 and b ¼ 0.4.

(a) Bifurcation curves on the parameter plane (α, v) related to the bifurcation values of map T<sup>Δ</sup> for a � c ¼ 2

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

Proposition 3. If we let v<sup>1</sup> ¼ v<sup>2</sup> ¼ v, the parameters a, b and c are fixed for

nized trajectories of the system (17) are divergent when ∀α ∈ 0; α<sup>0</sup> ½ Þ or ∀v∈ v<sup>0</sup> ð Þ ; þ∞ . In order to analyze the effect of any slight perturbation of one parameter on the

Then, the characteristic values of the Jacobian matrix J qð Þ ; q evaluated at any

λ<sup>k</sup> ¼ 1 þ vð Þ 1 � α ð Þþ a � c vqð Þ 10bα � 6b

where the corresponding eigenvectors are 1ð Þ ; 1 and 1ð Þ ; �1 , respectively. And

1 þ vð Þ 1 � α ð Þþ a � c vð Þ 10bα � 6b qi

1 þ vð Þ 1 � α ð Þþ a � c vð Þ 6bα � 4b qi

It is assumed that a period-k cycle f g ð Þ qð Þ1 ; qð Þ1 ;ð Þ qð Þ2 ; qð Þ2 ; ⋯;ð Þ q kð Þ; q kð Þ embedded into the invariant set Δ of the map T<sup>0</sup> is correspond to the cycle

the eigenvalue λ<sup>k</sup> is related to the invariant manifolds on the diagonal.

system, we study the transverse stability of an attractor A of map T<sup>0</sup>

Jacobian matrix of map T<sup>0</sup> on the diagonal can be obtained as follow,

1 þ vð Þ 1 � α ð Þþ a � c vqð Þ 8bα � 5b

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> of the speed of adjustment <sup>v</sup> does exist such that synchro-

!

vqð Þ 2bα � b

<sup>λ</sup><sup>⊥</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> vqð Þ <sup>6</sup>b<sup>α</sup> � <sup>4</sup><sup>b</sup> (24)

<sup>0</sup> when the synchronized phenomenon occurs, the

� � (25)

� � (26)

v að Þ �<sup>c</sup> of the weight of consumer surplus <sup>α</sup> or

1 þ vð Þ 1 � α ð Þþ a � c vqð Þ 8bα � 5b

. And the

(23)

derived,

Figure 4.

J qð Þ¼ ; q

51

a threshold <sup>v</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup>

system (17). Then, a threshold <sup>α</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> � <sup>3</sup>

DOI: http://dx.doi.org/10.5772/intechopen.87200

vqð Þ 2bα � b

point on the diagonal are given by,

f g qð Þ1 ; qð Þ2 ; ⋯; q kð Þ of the map T<sup>Δ</sup>

λ ð Þk <sup>k</sup> <sup>¼</sup> <sup>Y</sup> k

λ ð Þk <sup>⊥</sup> <sup>¼</sup> <sup>Y</sup> k

i¼1

i¼1

two multipliers are given as,

we can also prove that the map (21) is topologically conjugate to the standard logistic map y tð Þ¼ þ 1 μy tð Þð Þ 1 � y tð Þ , where

$$
\mu = \nu = 1 + \nu(1 - a)(a - c).
$$

Through the standard logistic map, we can easily analyze the dynamical behavior of the two-dimensional map T<sup>0</sup> on the diagonal Δ. Under this situation, the Nash equilibrium E<sup>∗</sup> of the system (10) is identical with the fixed point of map T<sup>Δ</sup> 0 . Since μ ¼ 1 þ vð Þ 1 � α ð Þ a � c , we take different values of the bifurcation parameter μ of the logistic map, and Figure 4a gives different bifurcation curves of the system on the parameter plane ð Þ α; v . The flip bifurcation occurs when the system parameter v equals <sup>v</sup> <sup>¼</sup> <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , and the Nash equilibrium <sup>E</sup><sup>∗</sup> loses its stability and forms a period-2 cycle around <sup>E</sup><sup>∗</sup> . At <sup>v</sup> <sup>¼</sup> ffiffi 6 p ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , the period-2 cycle generates a period-4 cycle after a flip bifurcation. When μ≈3:5699, the standard period doubling cascade ends and the system enters chaos. When v> <sup>3</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , the general trajectory of the map T<sup>Δ</sup> <sup>0</sup> is divergent.

As shown in Figure 4b, which the parameters is the same as Figure 4a, a twodimensional bifurcation diagram of the system with v and α is obtained. Since it has been proved that the map T<sup>Δ</sup> <sup>0</sup> is topologically conjugate to the logistic map, we can find that the bifurcation curves of the two graphs are the same. In Figure 4a, the curve <sup>C</sup><sup>1</sup> is correspond to the equation <sup>v</sup> <sup>¼</sup> <sup>2</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , and the region below it represents the set of points of v and α at 1<μ<3. In this region, the fix point is stable. That is, the synchronization trajectory converges to the Nash equilibrium point. In Figure 4b, it corresponds to the period-1 region below the green region. When the point above the curve C<sup>2</sup> passes through the curve C<sup>0</sup> in Figure 4a, the system goes into chaos through a period doubling cascade. In Figure 4a, the curve C<sup>∞</sup> is correspond to the equation <sup>v</sup> <sup>¼</sup> <sup>3</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , where the parameters <sup>v</sup> and <sup>α</sup> of the

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

Figure 4.

q1ð Þ¼ t þ 1 q1ðÞþt vq1ð Þt

on the diagonal Δ can be analyzed through the following map,

Similarly, through the following linear transformation

logistic map y tð Þ¼ þ 1 μy tð Þð Þ 1 � y tð Þ , where

period-2 cycle around <sup>E</sup><sup>∗</sup> . At <sup>v</sup> <sup>¼</sup> ffiffi

<sup>0</sup> is divergent.

curve <sup>C</sup><sup>1</sup> is correspond to the equation <sup>v</sup> <sup>¼</sup> <sup>2</sup>

been proved that the map T<sup>Δ</sup>

correspond to the equation <sup>v</sup> <sup>¼</sup> <sup>3</sup>

cascade ends and the system enters chaos. When v> <sup>3</sup>

the standard logistic map.

Research Advances in Chaos Theory

T<sup>Δ</sup>

equals <sup>v</sup> <sup>¼</sup> <sup>2</sup>

of the map T<sup>Δ</sup>

50

It is easy to verify that the dynamics on the axis q<sup>2</sup> is also controlled by the map (19). It means that the system (17) can be regarded as a 1-dimensional map at the coordinate axes. The map (19) is topologically conjugate to the standard logistic map x tð Þ¼ þ 1 ωx tð Þð Þ 1 � x tð Þ through a linear transformation, which is given as,

> <sup>q</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> vð Þ 2b � 3bα � �

and the parameter ω can be presented as ω ¼ 1 þ vð Þ 1 � α ð Þ a � c . Thus the nonlinear dynamics of system (17) on the invariant axes can be analyzed through

> <sup>q</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> vð Þ 3b � 5bα � �

we can also prove that the map (21) is topologically conjugate to the standard

μ ¼ ω ¼ 1 þ vð Þ 1 � α ð Þ a � c :

μ ¼ 1 þ vð Þ 1 � α ð Þ a � c , we take different values of the bifurcation parameter μ of the logistic map, and Figure 4a gives different bifurcation curves of the system on the parameter plane ð Þ α; v . The flip bifurcation occurs when the system parameter v

> 6 p

4 cycle after a flip bifurcation. When μ≈3:5699, the standard period doubling

Through the standard logistic map, we can easily analyze the dynamical behavior of the two-dimensional map T<sup>0</sup> on the diagonal Δ. Under this situation, the Nash equilibrium E<sup>∗</sup> of the system (10) is identical with the fixed point of map T<sup>Δ</sup>

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , and the Nash equilibrium <sup>E</sup><sup>∗</sup> loses its stability and forms a

As shown in Figure 4b, which the parameters is the same as Figure 4a, a twodimensional bifurcation diagram of the system with v and α is obtained. Since it has

find that the bifurcation curves of the two graphs are the same. In Figure 4a, the

sents the set of points of v and α at 1<μ<3. In this region, the fix point is stable. That is, the synchronization trajectory converges to the Nash equilibrium point. In Figure 4b, it corresponds to the period-1 region below the green region. When the point above the curve C<sup>2</sup> passes through the curve C<sup>0</sup> in Figure 4a, the system goes into chaos through a period doubling cascade. In Figure 4a, the curve C<sup>∞</sup> is

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , the period-2 cycle generates a period-

<sup>0</sup> is topologically conjugate to the logistic map, we can

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , where the parameters <sup>v</sup> and <sup>α</sup> of the

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , and the region below it repre-

It can also be proved that the diagonal Δ is an invariant set of system (17), i.e., the trajectory starting from the diagonal Δ will stay forever on it. Therefore, the dynamical behavior of system (17) can be analyzed through the map T<sup>0</sup> which is restricted to the diagonal. If we let q<sup>1</sup> ¼ q<sup>2</sup> ¼ q, then the dynamics generated by T<sup>0</sup>

<sup>0</sup> : q tð Þ¼ þ 1 q tð Þþ vq tð Þð Þ ð Þ 1 � α ð Þþ a � c ð Þ 5bα � 3b q tð Þ (21)

<sup>1</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þþ <sup>a</sup>�<sup>c</sup> ð Þ <sup>3</sup>bα�2<sup>b</sup> <sup>q</sup><sup>1</sup> ½ � ð Þ<sup>t</sup> (19)

x (20)

y (22)

ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> , the general trajectory

0 . Since

(a) Bifurcation curves on the parameter plane (α, v) related to the bifurcation values of map T<sup>Δ</sup> for a � c ¼ 2 and (b) 2D bifurcation diagram in the parameter plane (α, v) for a � c ¼ 2 and b ¼ 0.4.

upper area lead to the divergence of the trajectory, corresponding to the black area in Figure 4b. Through the above analysis, the following proposition can be derived,

Proposition 3. If we let v<sup>1</sup> ¼ v<sup>2</sup> ¼ v, the parameters a, b and c are fixed for system (17). Then, a threshold <sup>α</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> � <sup>3</sup> v að Þ �<sup>c</sup> of the weight of consumer surplus <sup>α</sup> or a threshold <sup>v</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup> ð Þ <sup>1</sup>�<sup>α</sup> ð Þ <sup>a</sup>�<sup>c</sup> of the speed of adjustment <sup>v</sup> does exist such that synchronized trajectories of the system (17) are divergent when ∀α ∈ 0; α<sup>0</sup> ½ Þ or ∀v∈ v<sup>0</sup> ð Þ ; þ∞ .

In order to analyze the effect of any slight perturbation of one parameter on the system, we study the transverse stability of an attractor A of map T<sup>0</sup> . And the Jacobian matrix of map T<sup>0</sup> on the diagonal can be obtained as follow,

$$J(q,q) = \begin{pmatrix} 1+v(\mathbf{1}-a)(a-c)+vq(8ba-5b) & vq(2ba-b) \\ vq(2ba-b) & 1+v(\mathbf{1}-a)(a-c)+vq(8ba-5b) \end{pmatrix} \tag{23}$$

Then, the characteristic values of the Jacobian matrix J qð Þ ; q evaluated at any point on the diagonal are given by,

$$\begin{aligned} \lambda\_{\parallel} &= \mathbf{1} + \nu(\mathbf{1} - a)(a - c) + \nu q(\mathbf{1}0ba - \mathbf{6}b) \\ \lambda\_{\perp} &= \mathbf{1} + \nu(\mathbf{1} - a)(a - c) + \nu q(\mathbf{6}ba - \mathbf{4}b) \end{aligned} \tag{24}$$

where the corresponding eigenvectors are 1ð Þ ; 1 and 1ð Þ ; �1 , respectively. And the eigenvalue λ<sup>k</sup> is related to the invariant manifolds on the diagonal.

It is assumed that a period-k cycle f g ð Þ qð Þ1 ; qð Þ1 ;ð Þ qð Þ2 ; qð Þ2 ; ⋯;ð Þ q kð Þ; q kð Þ embedded into the invariant set Δ of the map T<sup>0</sup> is correspond to the cycle f g qð Þ1 ; qð Þ2 ; ⋯; q kð Þ of the map T<sup>Δ</sup> <sup>0</sup> when the synchronized phenomenon occurs, the two multipliers are given as,

$$\lambda\_{\parallel}^{(k)} = \prod\_{i=1}^{k} \left( \mathbf{1} + v(\mathbf{1} - a)(a - c) + v(\mathbf{10}ba - \mathbf{6}b)q\_i \right) \tag{25}$$

$$\lambda\_{\perp}^{(k)} = \prod\_{i=1}^{k} \left( \mathbf{1} + v(\mathbf{1} - a)(a - c) + v(\mathbf{6}ba - 4b)q\_i \right) \tag{26}$$

Since the stability conditions of the period-k cycle on the diagonal Δ of system (17) is same with the one-dimensional map T<sup>Δ</sup> 0 , here we only study the transverse stability of the one-dimensional map T<sup>Δ</sup> 0 . Under this situation, the transverse eigenvalue evaluated at the Nash equilibrium point E<sup>∗</sup> is given by

$$
\lambda\_{\perp}^{E^\*} = \mathbf{1} + \frac{v(a-c)(\mathbf{1}-a)^2}{5a-3} \tag{27}
$$

Later, we will exhibit the attractors and their basins of attraction corresponding to different values v under this set of parameters, and analyze the changes of attractors

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

A closed invariant set A is a attractor which means that it is asymptotically

stable invariant set as attractor. A basin of attraction may contain one or more attractors that may coexist with a set of repel points that produce either intermittent chaos or a blurry boundary. The basin of attraction of attractor A is the set of

B Að Þ¼ <sup>q</sup>1ð Þ <sup>0</sup> ; <sup>q</sup>2ð Þ <sup>0</sup> <sup>T</sup><sup>m</sup> <sup>q</sup>1ð Þ <sup>0</sup> ; <sup>q</sup>2ð Þ <sup>0</sup> ! A as m ! <sup>∞</sup>

For the sake of analyzing the topological structure of the basin of attraction B Að Þ, we study the boundary of B Að Þ firstly. Suppose that the map T<sup>0</sup> has a unique attractor <sup>A</sup> at finite distance, let <sup>∂</sup>B Að Þ be the boundary of the basin B Að Þ, then it is also the boundary of the basin of infinity Bð Þ ∞ generated by unbounded trajectories. Firstly, we take the dynamics of system (17) into account and restrict it to the invariant axis. When vð Þ 1 � α ð Þ a � c < 3, if the initial conditions belong to the

the origin. It has been obtained previously that the dynamical behavior of system

(17) on the coordinate axis is governed by the map (19), so that 0<sup>i</sup>

:

, ið Þ <sup>¼</sup> <sup>1</sup>; <sup>2</sup> , according to map (17), we can obtain the bounded

<sup>q</sup>1ðÞþ<sup>t</sup> vq tð Þ ð Þ <sup>1</sup> � <sup>α</sup> ð Þþ <sup>a</sup> � <sup>c</sup> ð Þ <sup>3</sup>b<sup>α</sup> � <sup>2</sup><sup>b</sup> <sup>q</sup>1ð Þ<sup>t</sup> <sup>¼</sup> 0 (30)

�<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup>

Since <sup>ε</sup><sup>1</sup> and <sup>ε</sup><sup>2</sup> are the segments of the boundary <sup>∂</sup>B Að Þ, and <sup>∂</sup>B Að Þ is also the boundary of the basin of infinity <sup>B</sup>ð Þ <sup>∞</sup> , their rank-<sup>k</sup> preimages <sup>T</sup>0ð Þ �<sup>k</sup> ð Þ <sup>ε</sup><sup>i</sup> , ið Þ <sup>¼</sup> <sup>1</sup>; <sup>2</sup>

> q tðÞþ vq tð Þ½ð Þ 1 � α ð Þþ a � c ð Þ 3bα � 2b q tð Þ� ¼ p q tðÞþ vq tð Þ½ð Þ 1 � α ð Þþ a � c ð Þ 3bα � 2b q tð Þ� ¼ 0

> > , there are itself and O<sup>3</sup>

also belong to <sup>∂</sup>B Að Þ. We can compute the rank-<sup>1</sup> preimages of a point

We can easily obtain the rank-1 preimages of the origin, which are

vbð Þ <sup>5</sup><sup>α</sup> � <sup>3</sup> ;

Through the discussion above, we can get the following propositions,

�1

�<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup>

those initial conditions that cause the trajectory to converge to A, i.e.,

ð Þ U ⊆U and

�<sup>1</sup>, ið Þ ¼ 1; 2 is the rank-1 preimage of

vbð Þ <sup>3</sup><sup>α</sup> � <sup>2</sup> (31)

, i <sup>¼</sup> <sup>1</sup>, 2, according to

�<sup>1</sup> besides, i.e.,

(32)

�1

1 þ vð Þ 1 � α ð Þ a � c vbð Þ 5α � 3 (33)

�<sup>1</sup>, ið Þ ¼ 1; 2 can be

∈ A. We also define a asymptotically

and their basins of attraction when the parameter v varies.

stable, i.e., a neighborhood U of A does exist such that T<sup>0</sup>

4.3 Global bifurcation and basins of attraction

DOI: http://dx.doi.org/10.5772/intechopen.87200

! <sup>A</sup> when <sup>m</sup> ! <sup>∞</sup>, <sup>∀</sup> <sup>q</sup>1; <sup>q</sup><sup>2</sup>

<sup>T</sup>0ð Þ <sup>m</sup> <sup>q</sup>1; <sup>q</sup><sup>2</sup>

interval <sup>ε</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>i</sup>

�1

The result is given as,

the algebraic system as follows

�<sup>1</sup>; <sup>0</sup> and <sup>O</sup><sup>2</sup>

O1

53

�<sup>1</sup> <sup>¼</sup> 01

O3

trajectories along the invariant axes, where 0<sup>i</sup>

computed by the following algebraic system

01 �<sup>1</sup> <sup>¼</sup> <sup>0</sup><sup>2</sup>

<sup>P</sup> <sup>¼</sup> ð Þ <sup>p</sup>; <sup>0</sup> <sup>∈</sup>ε<sup>1</sup> or <sup>P</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>p</sup> <sup>∈</sup> <sup>ε</sup><sup>2</sup> that belongs to <sup>ε</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>i</sup>

�<sup>1</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>2</sup>

Through Eq. (27), we can draw the following conclusions directly. That is, when all the parameters satisfy 0 <sup>&</sup>lt;v að Þ � <sup>c</sup> ð Þ <sup>1</sup> � <sup>α</sup> <sup>2</sup> <sup>þ</sup> <sup>5</sup><sup>α</sup> <sup>&</sup>lt;3, the Nash equilibrium <sup>E</sup><sup>∗</sup> is transversely attractive.

As we know that an attractor A of T<sup>0</sup> is asymptotically stable if and only if all the trajectories that belong to attractor A are transversely attractive. To study the stability of the attractor, we can calculate its transverse Lyapunov exponent as,

$$\Lambda\_{\perp} = \lim\_{n \to \infty} \sum\_{i=0}^{n} \ln |\lambda\_{\perp}(q(i))| \tag{28}$$

where qð Þ 0 ∈ A and q ið Þ is a generic trajectory generated by the map TΔ. If the initial condition <sup>q</sup>ð Þ <sup>0</sup> belongs to a period-k cycle, then <sup>Λ</sup><sup>⊥</sup> <sup>¼</sup> ln <sup>λ</sup><sup>k</sup> ⊥ . In this case, if Λ<sup>⊥</sup> <0, then the period-k cycle is transversely stable. When the initial condition qð Þ 0 belongs to a generic aperiodic trajectory embedded in the chaotic attractors, then the transverse Lyapunov exponent Λ<sup>⊥</sup> is the natural transverse Lyapunov exponent Λnat <sup>⊥</sup> . Since many unstable cycles along the diagonal are embedded in the chaotic attractor A, a spectrum of transverse Lyapunov exponents can be determined by the inequality

$$
\Lambda\_{\perp}^{\min} \le \cdots \le \Lambda\_{\perp}^{\text{nat}} \le \cdots \le \Lambda\_{\perp}^{\max} \tag{29}
$$

If all cycles embedded in A are transversely stable (Λmax <sup>⊥</sup> < 0) then A is asymptotically stable in the Lyapunov sense. If some cycles embedded in the chaotic attractor A are transversely unstable (Λmax <sup>⊥</sup> >0 and Λnat <sup>⊥</sup> < 0) then A is not stable in the Lyapunov sense, but it is a stable Milnor attractor. So we can look for the Milnor attractors by transverse Lyapunov exponents.

Figure 5 gives the natural transverse Lyapunov exponent and the bifurcation diagram with the fixed parameter v when a � c ¼ 5:15, b ¼ 0:1911895 and α ¼ 0:4.

#### Figure 5.

Bifurcation diagram for the restriction of the map T<sup>Δ</sup> to the invariant diagonal and the corresponding transverse Lyapunov exponent for v∈[0.78, 0.91] given a � c ¼ 5.15, b ¼ 0.1911895 and α ¼ 0.4.

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

Later, we will exhibit the attractors and their basins of attraction corresponding to different values v under this set of parameters, and analyze the changes of attractors and their basins of attraction when the parameter v varies.

#### 4.3 Global bifurcation and basins of attraction

Since the stability conditions of the period-k cycle on the diagonal Δ of system

0

eigenvalue evaluated at the Nash equilibrium point E<sup>∗</sup> is given by

λE∗ <sup>⊥</sup> ¼ 1 þ 0

v að Þ � <sup>c</sup> ð Þ <sup>1</sup> � <sup>α</sup> <sup>2</sup>

Through Eq. (27), we can draw the following conclusions directly. That is, when all the parameters satisfy 0 <sup>&</sup>lt;v að Þ � <sup>c</sup> ð Þ <sup>1</sup> � <sup>α</sup> <sup>2</sup> <sup>þ</sup> <sup>5</sup><sup>α</sup> <sup>&</sup>lt;3, the Nash equilibrium <sup>E</sup><sup>∗</sup> is

As we know that an attractor A of T<sup>0</sup> is asymptotically stable if and only if all the

trajectories that belong to attractor A are transversely attractive. To study the stability of the attractor, we can calculate its transverse Lyapunov exponent as,

> n i¼0

where qð Þ 0 ∈ A and q ið Þ is a generic trajectory generated by the map TΔ. If the

<sup>⊥</sup> . Since many unstable cycles along the diagonal are embedded in the

<sup>⊥</sup> ≤⋯≤Λmax

<sup>⊥</sup> >0 and Λnat

Λ<sup>⊥</sup> <0, then the period-k cycle is transversely stable. When the initial condition qð Þ 0 belongs to a generic aperiodic trajectory embedded in the chaotic attractors, then the transverse Lyapunov exponent Λ<sup>⊥</sup> is the natural transverse Lyapunov

chaotic attractor A, a spectrum of transverse Lyapunov exponents can be deter-

<sup>⊥</sup> ≤⋯≤Λnat

totically stable in the Lyapunov sense. If some cycles embedded in the chaotic

the Lyapunov sense, but it is a stable Milnor attractor. So we can look for the Milnor

Figure 5 gives the natural transverse Lyapunov exponent and the bifurcation diagram with the fixed parameter v when a � c ¼ 5:15, b ¼ 0:1911895 and α ¼ 0:4.

Bifurcation diagram for the restriction of the map T<sup>Δ</sup> to the invariant diagonal and the corresponding transverse

Lyapunov exponent for v∈[0.78, 0.91] given a � c ¼ 5.15, b ¼ 0.1911895 and α ¼ 0.4.

<sup>Λ</sup><sup>⊥</sup> <sup>¼</sup> lim<sup>n</sup>!<sup>∞</sup> <sup>∑</sup>

initial condition <sup>q</sup>ð Þ <sup>0</sup> belongs to a period-k cycle, then <sup>Λ</sup><sup>⊥</sup> <sup>¼</sup> ln <sup>λ</sup><sup>k</sup>

Λmin

If all cycles embedded in A are transversely stable (Λmax

attractor A are transversely unstable (Λmax

attractors by transverse Lyapunov exponents.

, here we only study the transverse

<sup>5</sup><sup>α</sup> � <sup>3</sup> (27)

ln j j λ⊥ð Þ q ið Þ (28)

⊥ 

<sup>⊥</sup> (29)

<sup>⊥</sup> < 0) then A is asymp-

<sup>⊥</sup> < 0) then A is not stable in

. In this case, if

. Under this situation, the transverse

(17) is same with the one-dimensional map T<sup>Δ</sup>

stability of the one-dimensional map T<sup>Δ</sup>

Research Advances in Chaos Theory

transversely attractive.

exponent Λnat

Figure 5.

52

mined by the inequality

A closed invariant set A is a attractor which means that it is asymptotically stable, i.e., a neighborhood U of A does exist such that T<sup>0</sup> ð Þ U ⊆U and <sup>T</sup>0ð Þ <sup>m</sup> <sup>q</sup>1; <sup>q</sup><sup>2</sup> ! <sup>A</sup> when <sup>m</sup> ! <sup>∞</sup>, <sup>∀</sup> <sup>q</sup>1; <sup>q</sup><sup>2</sup> ∈ A. We also define a asymptotically stable invariant set as attractor. A basin of attraction may contain one or more attractors that may coexist with a set of repel points that produce either intermittent chaos or a blurry boundary. The basin of attraction of attractor A is the set of those initial conditions that cause the trajectory to converge to A, i.e.,

$$B(A) = \{ (q\_1(\mathbf{0}), q\_2(\mathbf{0})) \Big| T^m(q\_1(\mathbf{0}), q\_2(\mathbf{0})) \to A \text{ as } m \to \infty \}.$$

For the sake of analyzing the topological structure of the basin of attraction B Að Þ, we study the boundary of B Að Þ firstly. Suppose that the map T<sup>0</sup> has a unique attractor <sup>A</sup> at finite distance, let <sup>∂</sup>B Að Þ be the boundary of the basin B Að Þ, then it is also the boundary of the basin of infinity Bð Þ ∞ generated by unbounded trajectories. Firstly, we take the dynamics of system (17) into account and restrict it to the invariant axis. When vð Þ 1 � α ð Þ a � c < 3, if the initial conditions belong to the interval <sup>ε</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>i</sup> �1 , ið Þ <sup>¼</sup> <sup>1</sup>; <sup>2</sup> , according to map (17), we can obtain the bounded trajectories along the invariant axes, where 0<sup>i</sup> �<sup>1</sup>, ið Þ ¼ 1; 2 is the rank-1 preimage of the origin. It has been obtained previously that the dynamical behavior of system (17) on the coordinate axis is governed by the map (19), so that 0<sup>i</sup> �<sup>1</sup>, ið Þ ¼ 1; 2 can be computed by the following algebraic system

$$q\_1(t) + \nu q(t) \left[ (\mathbf{1} - a)(a - c) + (\Im b a - \Im b) q\_1(t) \right] = \mathbf{0} \tag{30}$$

The result is given as,

$$\mathbf{0}^1\_{-1} = \mathbf{0}^2\_{-1} = \frac{\mathbf{1} + v(\mathbf{1} - a)(a - c)}{vb(3a - 2)}\tag{31}$$

Since <sup>ε</sup><sup>1</sup> and <sup>ε</sup><sup>2</sup> are the segments of the boundary <sup>∂</sup>B Að Þ, and <sup>∂</sup>B Að Þ is also the boundary of the basin of infinity <sup>B</sup>ð Þ <sup>∞</sup> , their rank-<sup>k</sup> preimages <sup>T</sup>0ð Þ �<sup>k</sup> ð Þ <sup>ε</sup><sup>i</sup> , ið Þ <sup>¼</sup> <sup>1</sup>; <sup>2</sup> also belong to <sup>∂</sup>B Að Þ. We can compute the rank-<sup>1</sup> preimages of a point <sup>P</sup> <sup>¼</sup> ð Þ <sup>p</sup>; <sup>0</sup> <sup>∈</sup>ε<sup>1</sup> or <sup>P</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>p</sup> <sup>∈</sup> <sup>ε</sup><sup>2</sup> that belongs to <sup>ε</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>i</sup> �1 , i <sup>¼</sup> <sup>1</sup>, 2, according to the algebraic system as follows

$$\begin{cases} q(t) + \nu q(t)[(\mathbf{1} - a)(a - c) + (\mathbf{3}ba - \mathbf{2}b)q(t)] = p \\ q(t) + \nu q(t)[(\mathbf{1} - a)(a - c) + (\mathbf{3}ba - \mathbf{2}b)q(t)] = \mathbf{0} \end{cases} \tag{32}$$

We can easily obtain the rank-1 preimages of the origin, which are O1 �<sup>1</sup> <sup>¼</sup> 01 �<sup>1</sup>; <sup>0</sup> and <sup>O</sup><sup>2</sup> �<sup>1</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>2</sup> �1 , there are itself and O<sup>3</sup> �<sup>1</sup> besides, i.e.,

$$O\_{-1}^{3} = \left(\frac{\mathbf{1} + v(\mathbf{1} - a)(a - c)}{vb(5a - 3)}, \frac{\mathbf{1} + v(\mathbf{1} - a)(a - c)}{vb(5a - 3)}\right) \tag{33}$$

Through the discussion above, we can get the following propositions,

Proposition 4. Let 1<sup>&</sup>lt; <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> <sup>&</sup>lt; 3 and <sup>ε</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>i</sup> �1 , i <sup>¼</sup> <sup>1</sup>, 2 be the segments of the coordinate axes qi , i ¼ 1, 2, then we can obtain the boundary of B Að Þ as follow,

$$\partial B(A) = \left(\bigcup\_{k=0}^{\infty} T'^{(-k)}(\varepsilon\_1)\right) \cup \left(\bigcup\_{k=0}^{\infty} T'^{(-k)}(\varepsilon\_2)\right) \tag{34}$$

Basins of attraction may be connected or not. The connected basins of attraction are divided into simple connected and complex one, and the complex connected basins of attraction means the existence of holes. If A is a connected attractor, the direct basin of attraction D<sup>0</sup> of A is the largest connected area of the entire attractor domain D containing A. The system (17) has the coexistence of attractors in a set of given parameters, the basin of attraction D refers to the union of the domain of attraction of all attractors in such a situation.

Figure 6 shows the coexistence of attractors and their basins of attraction for given parameters a � c ¼ 5:3, b ¼ 0:234 and v ¼ 0:85. In Figure 6a, the parameter α is chosen as α ¼ 0:4, there are two attractors coexisting, one is a Milnor attractor A located on the diagonal and the other consisting of 4-piece chaos attractor is in symmetrical positions with respect to the diagonal, i.e., <sup>F</sup> <sup>¼</sup> <sup>∪</sup> <sup>4</sup> <sup>i</sup>¼<sup>1</sup>Fi. The basin of attraction is composed of the union of the attractive domain of two attractors. The attractive domain of the Milnor attractor A is the complex connected set, and the attraction domain of the attractor F is non-connected set. And the boundary of the 4-piece chaos attractor is just contact with the critical curve, and it is because of this contact that the system undergoes a global bifurcation. Figure 6b is the attractor and the attractive basin at α ¼ 0:387275 after the global bifurcation occurred in the Figure 6a. We can find 4 attractors coexisted in this figure the attractor F in Figure 6a undergoes a global bifurcation and turns into 3 period-4 cycles, and the attractor A is also a Milnor attractor. The attraction domain of the period-4 cycle is composed of some complex connected sets being in symmetrical positions with respect to the diagonal. As is shown in Figure 6b, there are many holes in the attracting domain, but it is a non-connected set. The basin of attraction of the Milnor attractor lying on the diagonal is still a complex connected set.

We have analyzed the global bifurcations that occur when the attractor's boundary contact to the critical curve, and we discuss the global bifurcation when the attractor contacts to the boundary of its basin of attraction. We also denote global bifurcation as "boundary crisis," the attractor is destroyed when it contacts to its basin of attraction. Figure 7 shows the coexistence of attractors and their basins

of attraction corresponding to the parameter a � c, when the parameters are chosen as α ¼ 0:5023335, b ¼ 0:4 and v<sup>1</sup> ¼ v<sup>2</sup> ¼ 0:85. We can see that as the difference between the maximum price a of a unit commodity and the marginal cost c increases, a period-4 cycle turns into 4-piece chaos attractor being in symmetrical positions with respect to the diagonal and finally merges into 2-piece chaos attractor. However, the 2-piece attractor being in symmetrical positions with respect to the diagonal grows larger as the parameter a � c increasing, until it contacts to its basin's boundary, and eventually occurs a global bifurcation, causing itself and its basin are destroyed until it disappears. We can also see its "ghost" in Figure 7d. This means that trajectories of the initial conditions that belong to the basin of attraction spend a long number of steps in the region occupied by the former attractor before converging to the other attractor. Figure 8 is the bifurcation diagram of the system

Basin of attractions for parameters b ¼ 0.4, α ¼ 0.502335 and v ¼ 0.85 (a) a � c ¼ 6, a two-piece chaotic attractor coexists with a period-4 cycle, (b) a � c ¼ 6.1, the 4-piece chaotic attractor coexist with a two-piece chaotic attractor, (c) a � c ¼ 6.1896, the 2-piece chaotic attractor formed by the a 4-piece chaotic attractor coexist with a two-piece chaotic attractor, and the many holes created by the global bifurcation and (d) a � c ¼

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

DOI: http://dx.doi.org/10.5772/intechopen.87200

6.26, a two-piece chaotic has a contact with its basin's boundary, and it is destroyed.

Figure 7.

55

at this set of parameters, and the bifurcation parameter is chosen as a � c.

can be restored clearly by numerical simulation method only. With the set of parameters in Figure 9 being identical to Figure 4, we select different speed of adjustment to analyze the change of attractors and their basins of attraction. We can observe that as the speed of adjustment changes from 0.79 to 0.9, the period-4 cycle being in symmetrical positions with respect to the diagonal of the system generates smooth limit cycle via a Neimark-Sacker bifurcation, as shown in Figure 9b, and the limit cycle becomes non-smooth gradually, and finally forms four-piece chaotic attractor, as shown in Figure 9d. The basin of attraction shrinks as the speed of adjustment v increasing. It is implied that when both firms choose a lower speed of

Figures 6 and 7 give two different global bifurcations, such bifurcations which

#### Figure 6.

Basin of attractions for parameters a � c ¼ 5.3, b ¼ 0.234 and v ¼ 0.85. (a) α ¼ 0.4, a four-cyclic chaotic attractor coexists with a Milnor attractor on the diagonal and (b) α ¼ 0.387275, a four-cyclic chaotic attractor has undergone a global bifurcation and three period-4 attractors coexist with a Milnor attractor.

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

#### Figure 7.

Proposition 4. Let 1<sup>&</sup>lt; <sup>v</sup>ð Þ <sup>1</sup> � <sup>α</sup> ð Þ <sup>a</sup> � <sup>c</sup> <sup>&</sup>lt; 3 and <sup>ε</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup><sup>i</sup>

∞ k¼0

symmetrical positions with respect to the diagonal, i.e., <sup>F</sup> <sup>¼</sup> <sup>∪</sup> <sup>4</sup>

Milnor attractor lying on the diagonal is still a complex connected set.

We have analyzed the global bifurcations that occur when the attractor's boundary contact to the critical curve, and we discuss the global bifurcation when the attractor contacts to the boundary of its basin of attraction. We also denote global bifurcation as "boundary crisis," the attractor is destroyed when it contacts to its basin of attraction. Figure 7 shows the coexistence of attractors and their basins

Basin of attractions for parameters a � c ¼ 5.3, b ¼ 0.234 and v ¼ 0.85. (a) α ¼ 0.4, a four-cyclic chaotic attractor coexists with a Milnor attractor on the diagonal and (b) α ¼ 0.387275, a four-cyclic chaotic attractor

has undergone a global bifurcation and three period-4 attractors coexist with a Milnor attractor.

<sup>T</sup>0ð Þ �<sup>k</sup> ð Þ <sup>ε</sup><sup>1</sup> 

Basins of attraction may be connected or not. The connected basins of attraction are divided into simple connected and complex one, and the complex connected basins of attraction means the existence of holes. If A is a connected attractor, the direct basin of attraction D<sup>0</sup> of A is the largest connected area of the entire attractor domain D containing A. The system (17) has the coexistence of attractors in a set of given parameters, the basin of attraction D refers to the union of the domain of

Figure 6 shows the coexistence of attractors and their basins of attraction for given parameters a � c ¼ 5:3, b ¼ 0:234 and v ¼ 0:85. In Figure 6a, the parameter α is chosen as α ¼ 0:4, there are two attractors coexisting, one is a Milnor attractor A located on the diagonal and the other consisting of 4-piece chaos attractor is in

attraction is composed of the union of the attractive domain of two attractors. The attractive domain of the Milnor attractor A is the complex connected set, and the attraction domain of the attractor F is non-connected set. And the boundary of the 4-piece chaos attractor is just contact with the critical curve, and it is because of this contact that the system undergoes a global bifurcation. Figure 6b is the attractor and the attractive basin at α ¼ 0:387275 after the global bifurcation occurred in the Figure 6a. We can find 4 attractors coexisted in this figure the attractor F in Figure 6a undergoes a global bifurcation and turns into 3 period-4 cycles, and the attractor A is also a Milnor attractor. The attraction domain of the period-4 cycle is composed of some complex connected sets being in symmetrical positions with respect to the diagonal. As is shown in Figure 6b, there are many holes in the attracting domain, but it is a non-connected set. The basin of attraction of the

<sup>∂</sup>B Að Þ¼ <sup>⋃</sup>

attraction of all attractors in such a situation.

of the coordinate axes qi

Research Advances in Chaos Theory

Figure 6.

54

�1

<sup>T</sup>0ð Þ �<sup>k</sup> ð Þ <sup>ε</sup><sup>2</sup> 

, i ¼ 1, 2, then we can obtain the boundary of B Að Þ as follow,

∪ ⋃∞ k¼0

, i <sup>¼</sup> <sup>1</sup>, 2 be the segments

<sup>i</sup>¼<sup>1</sup>Fi. The basin of

(34)

Basin of attractions for parameters b ¼ 0.4, α ¼ 0.502335 and v ¼ 0.85 (a) a � c ¼ 6, a two-piece chaotic attractor coexists with a period-4 cycle, (b) a � c ¼ 6.1, the 4-piece chaotic attractor coexist with a two-piece chaotic attractor, (c) a � c ¼ 6.1896, the 2-piece chaotic attractor formed by the a 4-piece chaotic attractor coexist with a two-piece chaotic attractor, and the many holes created by the global bifurcation and (d) a � c ¼ 6.26, a two-piece chaotic has a contact with its basin's boundary, and it is destroyed.

of attraction corresponding to the parameter a � c, when the parameters are chosen as α ¼ 0:5023335, b ¼ 0:4 and v<sup>1</sup> ¼ v<sup>2</sup> ¼ 0:85. We can see that as the difference between the maximum price a of a unit commodity and the marginal cost c increases, a period-4 cycle turns into 4-piece chaos attractor being in symmetrical positions with respect to the diagonal and finally merges into 2-piece chaos attractor. However, the 2-piece attractor being in symmetrical positions with respect to the diagonal grows larger as the parameter a � c increasing, until it contacts to its basin's boundary, and eventually occurs a global bifurcation, causing itself and its basin are destroyed until it disappears. We can also see its "ghost" in Figure 7d. This means that trajectories of the initial conditions that belong to the basin of attraction spend a long number of steps in the region occupied by the former attractor before converging to the other attractor. Figure 8 is the bifurcation diagram of the system at this set of parameters, and the bifurcation parameter is chosen as a � c.

Figures 6 and 7 give two different global bifurcations, such bifurcations which can be restored clearly by numerical simulation method only. With the set of parameters in Figure 9 being identical to Figure 4, we select different speed of adjustment to analyze the change of attractors and their basins of attraction. We can observe that as the speed of adjustment changes from 0.79 to 0.9, the period-4 cycle being in symmetrical positions with respect to the diagonal of the system generates smooth limit cycle via a Neimark-Sacker bifurcation, as shown in Figure 9b, and the limit cycle becomes non-smooth gradually, and finally forms four-piece chaotic attractor, as shown in Figure 9d. The basin of attraction shrinks as the speed of adjustment v increasing. It is implied that when both firms choose a lower speed of

adjustment, they can reach the balance easily in the game. However, the period-2 cycle embedded in the diagonal becomes a period-4 cycle, period-8 cycle, etc. That is, a flip-bifurcation happens. And finally a Milnor attractor forms with the increasing speed of adjustment. Its basin of attraction increases with the increasing speed of adjustment gradually. In the bifurcation diagram of Figure 5b, we can observe the process of entire bifurcation process.

4.4 Synchronization

DOI: http://dx.doi.org/10.5772/intechopen.87200

Figure 10.

57

In this section we study the formation mechanism of the synchronization trajectories. The trajectories starting from different initial conditions return to the diagonal eventually, i.e., <sup>q</sup>1ð Þ <sup>0</sup> 6¼ <sup>q</sup>2ð Þ <sup>0</sup> . A <sup>t</sup> <sup>∗</sup> does exist such that <sup>q</sup>1ðÞ¼ <sup>t</sup> <sup>q</sup>2ð Þ<sup>t</sup> when t>t <sup>∗</sup> , and we define such trajectories as synchronization. However, when the diagonal Δ is an invariant sub-manifold, synchronized dynamics occur. We have proved that the map T<sup>0</sup> can be obtained by two identical one-dimensional coupling

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

0 which is

<sup>k</sup>¼<sup>1</sup>T0ð Þ<sup>k</sup> ð Þ <sup>Γ</sup> .

maps, and the synchronization trajectory can be controlled by a map T<sup>Δ</sup>

topologically conjugate to the standard logistic map. When we choose the

Parameter values are chosen as v ¼ 1, a � c ¼ 5 and b ¼ 0.234. (a) Four-piece Milnor attractor of system T belonging to the diagonal for α ¼ 0.48365, (b) the displacement q1 � q2 versus time for the same parameters as in (a), (c) α ¼ 0.485092, a 16-cyclic chaotic attractor is in symmetrical positions with respect to the diagonal, (d) α ¼ 0.437609, a trajectory in the phase space (q1, q2) whose transient part is out of diagonal that synchronizes along the Milnor attractor in the long run, (e) α ¼ 0.476955, a two-cyclic chaotic attractor

coexists with a period-2 cycle and (f) boundary of the chaotic area obtained by <sup>∂</sup><sup>A</sup> <sup>¼</sup> <sup>∪</sup> <sup>6</sup>

Figure 8. One-dimensional bifurcation diagram with respect to a � c for the set of parameters in Figure 7.

#### Figure 9.

Basin of attractions for parameter a � c ¼ 5.15, b ¼ 0.1911895 and α ¼ 0.4. (a) v ¼ 0.79, an attracting four-period cycle coexists with the two-period cycle on the diagonal, (b) v ¼ 0.83, a period-4 cycle has undergone a Neimark-Sacker bifurcation and an attractor formed by four smooth curves coexist with a period-8 cycle, (c) v ¼ 0.872, a four-cyclic chaotic attractor formed by the four smooth curves coexists with the Milnor attractor and (d) v ¼ 0.9, a four-piece chaotic attractor exists outside the diagonal.

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

#### 4.4 Synchronization

adjustment, they can reach the balance easily in the game. However, the period-2 cycle embedded in the diagonal becomes a period-4 cycle, period-8 cycle, etc. That is, a flip-bifurcation happens. And finally a Milnor attractor forms with the increasing speed of adjustment. Its basin of attraction increases with the increasing speed of adjustment gradually. In the bifurcation diagram of Figure 5b, we can

One-dimensional bifurcation diagram with respect to a � c for the set of parameters in Figure 7.

Basin of attractions for parameter a � c ¼ 5.15, b ¼ 0.1911895 and α ¼ 0.4. (a) v ¼ 0.79, an attracting four-period cycle coexists with the two-period cycle on the diagonal, (b) v ¼ 0.83, a period-4 cycle has undergone a Neimark-Sacker bifurcation and an attractor formed by four smooth curves coexist with a period-8 cycle, (c) v ¼ 0.872, a four-cyclic chaotic attractor formed by the four smooth curves coexists with the

Milnor attractor and (d) v ¼ 0.9, a four-piece chaotic attractor exists outside the diagonal.

observe the process of entire bifurcation process.

Research Advances in Chaos Theory

Figure 8.

Figure 9.

56

In this section we study the formation mechanism of the synchronization trajectories. The trajectories starting from different initial conditions return to the diagonal eventually, i.e., <sup>q</sup>1ð Þ <sup>0</sup> 6¼ <sup>q</sup>2ð Þ <sup>0</sup> . A <sup>t</sup> <sup>∗</sup> does exist such that <sup>q</sup>1ðÞ¼ <sup>t</sup> <sup>q</sup>2ð Þ<sup>t</sup> when t>t <sup>∗</sup> , and we define such trajectories as synchronization. However, when the diagonal Δ is an invariant sub-manifold, synchronized dynamics occur. We have proved that the map T<sup>0</sup> can be obtained by two identical one-dimensional coupling maps, and the synchronization trajectory can be controlled by a map T<sup>Δ</sup> 0 which is topologically conjugate to the standard logistic map. When we choose the

#### Figure 10.

Parameter values are chosen as v ¼ 1, a � c ¼ 5 and b ¼ 0.234. (a) Four-piece Milnor attractor of system T belonging to the diagonal for α ¼ 0.48365, (b) the displacement q1 � q2 versus time for the same parameters as in (a), (c) α ¼ 0.485092, a 16-cyclic chaotic attractor is in symmetrical positions with respect to the diagonal, (d) α ¼ 0.437609, a trajectory in the phase space (q1, q2) whose transient part is out of diagonal that synchronizes along the Milnor attractor in the long run, (e) α ¼ 0.476955, a two-cyclic chaotic attractor coexists with a period-2 cycle and (f) boundary of the chaotic area obtained by <sup>∂</sup><sup>A</sup> <sup>¼</sup> <sup>∪</sup> <sup>6</sup> <sup>k</sup>¼<sup>1</sup>T0ð Þ<sup>k</sup> ð Þ <sup>Γ</sup> .

parameters as v ¼ 1, a � c ¼ 5 and b ¼ 0:234, the weight α varies, and we can observe that the dynamic behavior of system is controlled by the attractor on the diagonal in Figure 10. When α ¼ 0:4837015, we can observe a Milnor attractor in Figure 10a. This means that cycle embedded in the diagonal are transversely unstable and blowout phenomenon occurs when the trajectory is near diagonal. The trajectory converges to the unique Milnor attractor embedded in the diagonal after experiencing a long transient. Figure 10b shows that the evolution of q<sup>1</sup> � q<sup>2</sup> versus time and synchronization is observed after a long transient. This is a typical on-off intermittency phenomenon. We can observe 16-piece chaos attractor being in symmetrical positions with respect to the diagonal in Figure 10c, when α increases to 0:485092. Figure 10d shows a chaotic attractor when α decreases to 0:437609, then synchronization occurs. As shown in Figure 10f, we adopt the trial-and-error method, with suitable part of LC�<sup>1</sup> taken as the starting part of Γ ¼ A∩LC�<sup>1</sup> to obtain the boundary of the chaotic attractor A and the entire basin of attraction in Figure 10e, i.e., the boundary of the chaotic attractor <sup>A</sup> is <sup>∂</sup><sup>A</sup> <sup>¼</sup> <sup>∪</sup> <sup>6</sup> <sup>k</sup>¼<sup>1</sup>T0ð Þ<sup>k</sup> ð Þ <sup>Γ</sup> .

of the attractor non connected. In addition, if we fixed parameters of the system, and change the values of the parameters a c only, we find another global bifurcation called "boundary crisis," i.e., when the attractor contact with its boundary of the basin of attraction, one of the attractors and its basin of attraction will be

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

destroyed.

DOI: http://dx.doi.org/10.5772/intechopen.87200

Author details

\* and Tong Chu2

provided the original work is properly cited.

\*Address all correspondence to: wei\_zhou@vip.126.com

1 School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, China

2 School of Law, Zhejiang University of Finance and Economics, Hangzhou, China

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Wei Zhou<sup>1</sup>

59

#### 5. Conclusion

In this chapter, the nonlinear dynamics of a Cournot duopoly game with bounded rationality is investigated. Unlike the existing literature, we suppose that the two firms not only pursue profit maximization but also take consumer surplus into account. Meanwhile, the objection of firms is supposed as the weighted sum of profit and consumer surplus. Based on the theory of gradient adjustment, all the firms adjust the output of next period according to the estimation of "marginal goal." The existence and stability of fixed points are analyzed. It is found that the boundary equilibrium point is always unstable, no matter what the parameters of the system are satisfied. At the same time, with the two-dimensional bifurcation diagram as the tool, the stability of the Nash equilibrium is analyzed. We found that the Nash equilibrium will lose its stability when the speed of adjustment of firms is too large, which maybe lead the market into chaos. The stability region of the Nash equilibrium will be only affected by the weight of consumer surplus. And the parameters a � c and b have hardly effect on the stability region of the Nash equilibrium. Meanwhile, we found that the two-dimensional bifurcation diagram have a beautiful fractal structure, but there are also many scattered points which is due to the coexistence of multiple attractors of the system through numerical simulation. By selecting corresponding parameters in the two-dimensional bifurcation diagram with scattered points, we draw the corresponding basin of attraction, and found the model not only has two attractors coexistence phenomenon, but also has 3, or even 4 attractors coexistence phenomenon.

Moreover, with the theory of invertible mapping and the critical curves of the system, the topological structure of basin of attraction is analyzed. By calculating the transverse Lyapunov exponent, the weak chaotic attractor of the system in the sense of Milnor is found, and the synchronization of the system is further studied. If we fix the other parameters of the system, and only change the weight of the firm to the consumer surplus, we can find on-off intermittency phenomenon and synchronization phenomenon. With the increasing of α, the synchronization phenomenon is vanished and a 16-piece chaotic attractor being in symmetrical position with respect to the main diagonal is produced. Under another set of parameters, and the parameter α is chosen as the bifurcation parameter. Through numerical simulation, it can be found that when the critical curve contact with the boundary of the basin, a global bifurcation is obtained. The global bifurcation makes the basin of attraction Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus DOI: http://dx.doi.org/10.5772/intechopen.87200

of the attractor non connected. In addition, if we fixed parameters of the system, and change the values of the parameters a c only, we find another global bifurcation called "boundary crisis," i.e., when the attractor contact with its boundary of the basin of attraction, one of the attractors and its basin of attraction will be destroyed.

#### Author details

parameters as v ¼ 1, a � c ¼ 5 and b ¼ 0:234, the weight α varies, and we can observe that the dynamic behavior of system is controlled by the attractor on the diagonal in Figure 10. When α ¼ 0:4837015, we can observe a Milnor attractor in Figure 10a. This means that cycle embedded in the diagonal are transversely unstable and blowout phenomenon occurs when the trajectory is near diagonal. The trajectory converges to the unique Milnor attractor embedded in the diagonal after experiencing a long transient. Figure 10b shows that the evolution of q<sup>1</sup> � q<sup>2</sup> versus time and synchronization is observed after a long transient. This is a typical on-off intermittency phenomenon. We can observe 16-piece chaos attractor being in symmetrical positions with respect to the diagonal in Figure 10c, when α increases to 0:485092. Figure 10d shows a chaotic attractor when α decreases to 0:437609, then synchronization occurs. As shown in Figure 10f, we adopt the trial-and-error method, with suitable part of LC�<sup>1</sup> taken as the starting part of Γ ¼ A∩LC�<sup>1</sup> to obtain the boundary of the chaotic attractor A and the entire basin of attraction in

Figure 10e, i.e., the boundary of the chaotic attractor <sup>A</sup> is <sup>∂</sup><sup>A</sup> <sup>¼</sup> <sup>∪</sup> <sup>6</sup>

has 3, or even 4 attractors coexistence phenomenon.

In this chapter, the nonlinear dynamics of a Cournot duopoly game with bounded rationality is investigated. Unlike the existing literature, we suppose that the two firms not only pursue profit maximization but also take consumer surplus into account. Meanwhile, the objection of firms is supposed as the weighted sum of profit and consumer surplus. Based on the theory of gradient adjustment, all the firms adjust the output of next period according to the estimation of "marginal goal." The existence and stability of fixed points are analyzed. It is found that the boundary equilibrium point is always unstable, no matter what the parameters of the system are satisfied. At the same time, with the two-dimensional bifurcation diagram as the tool, the stability of the Nash equilibrium is analyzed. We found that the Nash equilibrium will lose its stability when the speed of adjustment of firms is too large, which maybe lead the market into chaos. The stability region of the Nash equilibrium will be only affected by the weight of consumer surplus. And the parameters a � c and b have hardly effect on the stability region of the Nash equilibrium. Meanwhile, we found that the two-dimensional bifurcation diagram have a beautiful fractal structure, but there are also many scattered points which is due to the coexistence of multiple attractors of the system through numerical simulation. By selecting corresponding parameters in the two-dimensional bifurcation diagram with scattered points, we draw the corresponding basin of attraction, and found the model not only has two attractors coexistence phenomenon, but also

Moreover, with the theory of invertible mapping and the critical curves of the system, the topological structure of basin of attraction is analyzed. By calculating the transverse Lyapunov exponent, the weak chaotic attractor of the system in the sense of Milnor is found, and the synchronization of the system is further studied. If we fix the other parameters of the system, and only change the weight of the firm to the consumer surplus, we can find on-off intermittency phenomenon and synchronization phenomenon. With the increasing of α, the synchronization phenomenon is vanished and a 16-piece chaotic attractor being in symmetrical position with respect to the main diagonal is produced. Under another set of parameters, and the parameter α is chosen as the bifurcation parameter. Through numerical simulation, it can be found that when the critical curve contact with the boundary of the basin, a global bifurcation is obtained. The global bifurcation makes the basin of attraction

5. Conclusion

Research Advances in Chaos Theory

58

<sup>k</sup>¼<sup>1</sup>T0ð Þ<sup>k</sup> ð Þ <sup>Γ</sup> .

Wei Zhou<sup>1</sup> \* and Tong Chu2

1 School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, China

2 School of Law, Zhejiang University of Finance and Economics, Hangzhou, China

\*Address all correspondence to: wei\_zhou@vip.126.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Friedman J. Oligopoly Theory. London: Cambridge University Press; 1983

[2] Elsadany AA. Dynamics of cournot duopoly game with bounded rationality based on relatve profit maximization. Applied Mathematics and Computation. 2017;294:253-263. DOI: 10.1016/j. amc.2016.09.018

[3] Andaluz J, Jarne G. Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational. Annals of Operations Research. 2016;238:1-25. DOI: 10.1007/s10479-015-2057-4

[4] Ma J, Guo Z. The influence of information on the stability of a dynamic Bertrand game. Communications in Nonlinear Science and Numerical Simulation. 2016;30:32-44. DOI: 10.1016/j.cnsns.2015.06.004

[5] Matsumoto A, Szidarovszky F, Yoshida H. Dynamics in linear Cournot duopolies with two time delays. Computational Economics. 2011;38:311. DOI: 10.1007/s10614-011-9295-6

[6] Gori L, Pecora N, Sodini M. Market share delegation in a nonlinear duopoly with quantity competition: The role of dynamic entry barriers. Journal of Evolutionary Economics. 2017;27:905-931

[7] Andaluz J, Jarne G. On the dynamics of economic games based on product differentiation. Mathematics and Computers in Simulation. 2015;113:16-27. DOI: 10.1016/j.matcom.2015.02.005

[8] Zhang Y, Zhou W, Chu T, et al. Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production. Nonlinear Dynamics. 2018;91:819-835. DOI: 10.1007/s11071-017-3912-4

[9] Zhou J, Zhou W, Chu T, Chang YX, Huang MJ. Bifurcation, intermittent

chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation. Applied Mathematics and Computation. 2019; 341:358-378. DOI: 10.1016/j. amc.2018.09.004

Chapter 4

Abstract

1. Introduction

61

Nonlinear Dynamical Regimes

and Control of Turbulence

Ginzburg-Landau Equation

Joël Bruno Gonpe Tafo, Laurent Nana, Conrad Bertrand Tabi

The dynamical behavior of pulse and traveling hole in a one-dimensional system depending on the boundary conditions, obeying the complex Ginzburg-Landau (CGL) equation, is studied numerically using parameters near a subcritical bifurcation. In a spatially extended system, the criterion of Benjamin-Feir-Newell (BFN) instability near the weakly inverted bifurcation is established, and many types of regimes such as laminar regime, spatiotemporal regime, defect turbulence regimes, and so on are observed. In finite system by using the homogeneous boundary conditions, two types of regimes are detected mainly the convective and the absolute instability. The convectively unstable regime appears below the threshold of the parameter control, and beyond, the absolute regime is observed. Controlling such regimes remains a great challenge; many methods such as the nonlinear diffusion parameter control are used. The unstable traveling hole in the onedimensional cubic-quintic CGL equation may be effectively stabilized in the chaotic regime. In order to stabilize defect turbulence regimes, we use the global time-delay auto-synchronization control; we also use another method of control which consists in modifying the nonlinear diffusion term. Finally, we control the unstable regimes by adding the nonlinear gradient term to the system. We then notice that the

through the Complex

chaotic system becomes stable under strong nonlinearity.

Ginzburg-Landau equation, unstable traveling hole

Keywords: Benjamin-Feir-Newell instability, subcritical bifurcation, complex

Many complex systems evolve in a non-equilibrium environment. Further out of the equilibrium [1], these systems tend to display progressively more complicated dynamics. The non-chaotic patterned state and spatiotemporal chaos are observed in the system. In the domain of the envelope equations, the quintic complex Ginzburg-Landau (CGL) equation is appropriate to obtain stable localized solutions (pulses, holes) [1, 2]. Among physical applications of the quintic CGL equation, one

and Timoléon Crépin Kofané

[10] De Fraja G, Delbono F. Alternative strategies of a public enterprise in oligopoly. Oxford Economic Papers. 1989;41:302-311. DOI: 10.1093/ oxfordjournals.oep.a041896

[11] Matsumura T. Partial privatization in mixed duopoly. Journal of Public Economics. 1998;70:473-483. DOI: 10.1016/S0047-2727(98)00051-6

[12] Fujiwara K. Partial privatization in a differentiated mixed oligopoly. Journal of Economics. 2007;92:51-65. DOI: 10.1007/s00712-007-0267-1

[13] Elsadany AA, Awad AM. Nonlinear dynamics of Cournot duopoly game with social welfare. Electronic Journal of Mathematical Analysis and Applications. 2016;4:173-191

[14] Dixit AK. A model of duopoly suggesting a theory of entry barriers. Bell Journal of Economics. 1979;10: 20-32. DOI: 10.2307/3003317

[15] Singh N, Vives X. Price and quantity competition in a differentiated duopoly. RAND Journal of Economics. 1984;15: 546-554. DOI: 10.2307/2555525

[16] Bischi GI, Sbragia L, Szidarovszky F. Learning the demand function in a repeated Cournot oligopoly game. International Journal of Systems Science. 2008;39:403-419. DOI: 10.1080/00207720701792131

[17] Fanti L, Gori L, Sodini M. Nonlinear dynamics in a Cournot duopoly with relative profit delegation. Chaos, Solitons & Fractals. 2012;45:1469-1478. DOI: 10.1016/j.chaos.2012.08.008

#### Chapter 4

References

amc.2016.09.018

1983

[1] Friedman J. Oligopoly Theory. London: Cambridge University Press;

Research Advances in Chaos Theory

[2] Elsadany AA. Dynamics of cournot duopoly game with bounded rationality based on relatve profit maximization. Applied Mathematics and Computation. 2017;294:253-263. DOI: 10.1016/j.

chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation. Applied Mathematics and Computation. 2019;

[10] De Fraja G, Delbono F. Alternative strategies of a public enterprise in oligopoly. Oxford Economic Papers. 1989;41:302-311. DOI: 10.1093/ oxfordjournals.oep.a041896

[11] Matsumura T. Partial privatization in mixed duopoly. Journal of Public Economics. 1998;70:473-483. DOI: 10.1016/S0047-2727(98)00051-6

[12] Fujiwara K. Partial privatization in a differentiated mixed oligopoly. Journal of Economics. 2007;92:51-65. DOI: 10.1007/s00712-007-0267-1

[13] Elsadany AA, Awad AM. Nonlinear dynamics of Cournot duopoly game with social welfare. Electronic Journal of

Mathematical Analysis and Applications. 2016;4:173-191

[14] Dixit AK. A model of duopoly suggesting a theory of entry barriers. Bell Journal of Economics. 1979;10: 20-32. DOI: 10.2307/3003317

[15] Singh N, Vives X. Price and quantity competition in a differentiated duopoly. RAND Journal of Economics. 1984;15: 546-554. DOI: 10.2307/2555525

[16] Bischi GI, Sbragia L, Szidarovszky F. Learning the demand function in a repeated Cournot oligopoly game. International Journal of Systems Science. 2008;39:403-419. DOI: 10.1080/00207720701792131

[17] Fanti L, Gori L, Sodini M. Nonlinear dynamics in a Cournot duopoly with relative profit delegation. Chaos, Solitons & Fractals. 2012;45:1469-1478. DOI: 10.1016/j.chaos.2012.08.008

341:358-378. DOI: 10.1016/j.

amc.2018.09.004

[3] Andaluz J, Jarne G. Stability of vertically differentiated Cournot and Bertrand-type models when firms are

boundedly rational. Annals of Operations Research. 2016;238:1-25. DOI: 10.1007/s10479-015-2057-4

[4] Ma J, Guo Z. The influence of

[5] Matsumoto A, Szidarovszky F, Yoshida H. Dynamics in linear Cournot

duopolies with two time delays.

Computational Economics. 2011;38:311. DOI: 10.1007/s10614-011-9295-6

[6] Gori L, Pecora N, Sodini M. Market share delegation in a nonlinear duopoly with quantity competition: The role of dynamic entry barriers. Journal of

Evolutionary Economics. 2017;27:905-931

[7] Andaluz J, Jarne G. On the dynamics of economic games based on product differentiation. Mathematics and Computers in Simulation. 2015;113:16-27. DOI: 10.1016/j.matcom.2015.02.005

[8] Zhang Y, Zhou W, Chu T, et al. Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production. Nonlinear Dynamics. 2018;91:819-835. DOI: 10.1007/s11071-017-3912-4

[9] Zhou J, Zhou W, Chu T, Chang YX, Huang MJ. Bifurcation, intermittent

60

information on the stability of a dynamic Bertrand game. Communications in Nonlinear Science and Numerical Simulation. 2016;30:32-44. DOI: 10.1016/j.cnsns.2015.06.004
