**1. Introduction**

It is well known that the wave propagation depends mainly on its velocity and frequency in one direction for a single wave[1-3]. There are many literatures devoted researches of single wave propagation such as solitary wave, periodic wave, chirped wave, rational wave etc [4-6]. However, what can be happen when two and even more waves with different features propagate together along different directions? In the past decades, many methods have been proposed for seeking two waves and multi-wave solutions to nonlinear models in modern physics. Recently, some effective and straight methods have been proposed such as homoclinic test approach(HTA)[7-8], extended homoclinic test approach(EHTA)[9-10] and three wave method [11-12]. These methods were applied to many nonlinear models. Several exact waves with different properties have been found out, such as periodic solitary wave, breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave, kinky kink wave, periodic kink wave, two-solitary wave, doubly periodic wave, doubly breather solitary wave, and so on. Because of interaction between waves with different features in propagation process of two-wave or multi-wave, some new phenomena have been discovered and numerically simulated, for example, resonance and non resonance, fission and fusion, bifurcation and deflexion etc. Furthermore, similar to the bifurcation theory of differential dynamical system, constant equilibrium solution of nonlinear evolution equation and propagation velocity of a wave as parameters are introduced to original equation, and then by using the small perturbation of parameter at a special value, two-wave or multi-wave propagation occurs new spatiotemporal change such as bifurcation of breather multi-soliton, periodic bifurcation and soliton degeneracy and so on.

This chapter mainly focus on explanation of different test methods and comprehensive applications to two-wave or multi-wave propagation. New methods will be described such as HTA, EHTA, Three-wave method and parameter small perturbation method. The spatiotemporal variety in exact two-wave and multi-wave propagation will be investigated and numerically simulated. In this chapter, some important models such as shallow water wave propagation models under the transverse long-wave disturbance Potential

©2012 Dai et al., licensee InTech. This is an open access chapter 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. © 2012 The Author(s). Licensee InTech. 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.

Kadomtsev-Petviashvili equation [13-17], Kadomtsev-Petviashvili [18-27] equation, and specially, Kadomtsev-Petviashvili with positive dispersion equation are considered. Applied new methods to these equations, some new two-wave and multi-wave are obtained and spatiotemporal variety in multi-wave propagation is investigated and numerically simulated.

This chapter consists of four sections. In section 1, we introduce some methods including homoclinic test approach (HTA), extended homoclinic test approach(EHTA), three-wave method and parameter small perturbation method. In section 2, we consider the potential Kadomtsev-Petviashvili equation and investigate the exact periodic kink-wave and degeneracy of soliton. In section 3, we consider the Kadomtsev-Petviashvili equation and investigate periodic bifurcation, deflexion of two solitary waves. In section 4, we consider the Kadomtsev-Petviashvili equation with positive dispersion. By using three-wave method, we obtain some breather kinds of multi-solitary wave solutions, and investigate the fission and fusion of multi-wave.

## **2. Some methods for seeking two-wave and multi-wave**

## **2.1. Homoclinic test approach**

Consider a (2+1) dimensional nonlinear evolution equation of the general form

$$F(\mathfrak{u}, \mathfrak{u}\_{l\prime} \mathfrak{u}\_{\mathfrak{x}\prime} \mathfrak{u}\_{\mathfrak{y}\prime} \cdots) = 0 \tag{1}$$

where all of *d*, *kj*, *lj*, *cj*, *j* = 1, 2 may be real numbers or complex numbers. This process is similar to the procedure that one can obtain the homoclinic orbit of the defocusing nonlinear Schrödinger equation from its dark-hole soliton solutions[1]. As a result, we call this skill as

To derive analytic expression, we can take the following procedure in detail: inserting Eq.(6) into Eq.(2), then equating the coefficients of the same kind terms to zero, subsequently, solving the resulting algebraic equations to determine the relationship between variables *kj*, *lj*, *j* = 1, 2, ··· with the help of symbolic computation software such as Maple. In Eq.(7), noting the cos functions is meaningful because we often take into account periodic effect in real physical background. Indeed, we can observe this solution is periodic breathing from their profile.

After substituting Eq.(6) into Eq.(2), we can get that whether the Eq.(2) has the nonzero solution. Furthermore, we do some mathematical simplicity. Rewrite Eq.(6) as follows:

√

√

where *b*1, *d* are constants. Now, factoring out the exponentials produces: Exploiting this Ans*atz* ¨ to obtain the new solutions of nonlinear evolution equation is called "Extended Homoclinic test approach". To derive analytic expression, we can take the following procedure in detail: inserting Eq.(10) into Eq.(2), then equating the coefficients of the same kind terms to zero, subsequently, solving the resulting algebraic equations to determine the relationship between variables *kj*, *lj*, *j* = 1, 2, ··· with the help of symbolic computation software such as

Multi-wave solutions are important because they reveal the interactions between the inner-waves and the various frequency and velocity components. The whole multi-wave solution, for instance, may sometimes be converted into a single soliton of very high energy that propagates over large regions of space without dispersing and an extremely destructive wave is therefore produced of which the tsunami is a good example. Since all double-wave solutions can be found by using the exp-function method proposed by Fu and Dai [18], we propose an extension of the three-soliton method [6] in this section (called the three-wave method) for finding coupled wave solutions. Consider a high dimensional

where *u* = *u*(*x*, *y*, *z*, *t*) and *F* is a polynomial *u* of and its derivatives, *t* represents time variable

and *x*, *y*, *z* represent spatial variables. The three-wave method operates as follows.

<sup>√</sup>*<sup>d</sup>* cosh(*η*<sup>2</sup> <sup>+</sup> *ln*(

*<sup>f</sup>* <sup>=</sup> <sup>√</sup>*<sup>d</sup>* cosh(*η*<sup>2</sup> <sup>+</sup> *ln*(

*f* = *eη*<sup>2</sup> (*e*−*η*<sup>2</sup> + cos(*η*1) + *deη*<sup>2</sup> ) (8)

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 107

*F*(*u*, *ut*, *ux*, *uy*, *uz*, *uxx*, ···) = 0 (11)

*u* = *T*(*f*) (12)

*d*)) + cos(*η*1)] (9)

*d*)) + *b*<sup>1</sup> cos(*η*1) (10)

"Homoclinic test approach".

or

Maple.

We replace Eq.(9) with

**2.3. Three-wave method**

nonlinear evolution equation of the general form

Step 1: By Painleve analysis, a transformation

**2.2. Extended homoclinic test approach**

*f* = *eη*<sup>2</sup> [

where *F* is a polynomial of *u*(*x*, *y*, *t*) and its derivatives, *t* represents time variable and *x*, *y* represent spatial variables. Assume that there exists a transformation of unknown function such that Eq.(1) becomes a bilinear equation in the following form

$$G(D\_{t\prime}D\_{\mathbf{x}\prime}D\_{y\prime}\cdots)f\cdot f = 0\tag{2}$$

where *G* is a general polynomial in *Dt*, *Dx*, *Dy*, ··· , where the Hirota's bilinear operator D-operator is defined by[2]

$$D\_{\mathbf{x}}^{\mathfrak{m}}D\_{t}^{\mathfrak{n}}f(\mathbf{x},t)\cdot\mathbf{g}(\mathbf{x},t)=(\frac{\partial}{\partial\mathbf{x}}-\frac{\partial}{\partial\mathbf{x}'})^{\mathfrak{m}}(\frac{\partial}{\partial t}-\frac{\partial}{\partial t'})^{\mathfrak{n}}[f(\mathbf{x},t)\mathbf{g}(\mathbf{x}',t')]|\_{\mathbf{x}'=\mathbf{x},t'=t}\tag{3}$$

Traditionally, one obtains two solition solutions with the assumption

$$f = 1 + e^{\eta\_1} + e^{\eta\_2} + de^{\eta\_1 + \eta\_2} \tag{4}$$

where *d* is a constant, *η*<sup>1</sup> = *k*1*x* + *l*1*y* + *c*1*t*, *η*<sup>2</sup> = *k*2*x* + *l*2*y* + *c*2*t*, and *kj*, *lj*, *cj*, *j* = 1, 2 are real numbers. If we treat *k*<sup>1</sup> and *k*<sup>2</sup> as complex numbers by taking *k*<sup>1</sup> = −*k*<sup>2</sup> = *ik*, *i* <sup>2</sup> <sup>=</sup> <sup>−</sup>1 and assuming *l*<sup>1</sup> = *l*<sup>2</sup> = *l* and *c*<sup>1</sup> = *c*<sup>2</sup> = *c*, Eq.(4) can be convert into the following form

$$f = 1 + \cos(k\mathbf{x})e^{ly + ct} + de^{2ly + 2ct} \tag{5}$$

In order to get more forms solution of Eq.(1), we use a more general Ans*atz* ¨ which contains complete variables of Eq.(2) replacing Eq.(5):

$$f = 1 + \cos(\eta\_1)e^{\eta\_2} + de^{2\eta\_2} \tag{6}$$

where

$$\begin{cases} \eta\_1 = k\_1 \mathfrak{x} + l\_1 \mathfrak{y} + c\_1 t \\ \eta\_2 = k\_2 \mathfrak{x} + l\_2 \mathfrak{y} + c\_2 t \end{cases} \tag{7}$$

where all of *d*, *kj*, *lj*, *cj*, *j* = 1, 2 may be real numbers or complex numbers. This process is similar to the procedure that one can obtain the homoclinic orbit of the defocusing nonlinear Schrödinger equation from its dark-hole soliton solutions[1]. As a result, we call this skill as "Homoclinic test approach".

To derive analytic expression, we can take the following procedure in detail: inserting Eq.(6) into Eq.(2), then equating the coefficients of the same kind terms to zero, subsequently, solving the resulting algebraic equations to determine the relationship between variables *kj*, *lj*, *j* = 1, 2, ··· with the help of symbolic computation software such as Maple. In Eq.(7), noting the cos functions is meaningful because we often take into account periodic effect in real physical background. Indeed, we can observe this solution is periodic breathing from their profile.

#### **2.2. Extended homoclinic test approach**

After substituting Eq.(6) into Eq.(2), we can get that whether the Eq.(2) has the nonzero solution. Furthermore, we do some mathematical simplicity. Rewrite Eq.(6) as follows:

$$f = e^{\eta\_2} \left( e^{-\eta\_2} + \cos(\eta\_1) + de^{\eta\_2} \right) \tag{8}$$

or

2 Will-be-set-by-IN-TECH

Kadomtsev-Petviashvili equation [13-17], Kadomtsev-Petviashvili [18-27] equation, and specially, Kadomtsev-Petviashvili with positive dispersion equation are considered. Applied new methods to these equations, some new two-wave and multi-wave are obtained and spatiotemporal variety in multi-wave propagation is investigated and numerically simulated. This chapter consists of four sections. In section 1, we introduce some methods including homoclinic test approach (HTA), extended homoclinic test approach(EHTA), three-wave method and parameter small perturbation method. In section 2, we consider the potential Kadomtsev-Petviashvili equation and investigate the exact periodic kink-wave and degeneracy of soliton. In section 3, we consider the Kadomtsev-Petviashvili equation and investigate periodic bifurcation, deflexion of two solitary waves. In section 4, we consider the Kadomtsev-Petviashvili equation with positive dispersion. By using three-wave method, we obtain some breather kinds of multi-solitary wave solutions, and investigate the fission and

**2. Some methods for seeking two-wave and multi-wave**

such that Eq.(1) becomes a bilinear equation in the following form

Traditionally, one obtains two solition solutions with the assumption

*<sup>t</sup> <sup>f</sup>*(*x*,*t*) · *<sup>g</sup>*(*x*, *<sup>t</sup>*)=( *<sup>∂</sup>*

complete variables of Eq.(2) replacing Eq.(5):

Consider a (2+1) dimensional nonlinear evolution equation of the general form

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>∂</sup>*

numbers. If we treat *k*<sup>1</sup> and *k*<sup>2</sup> as complex numbers by taking *k*<sup>1</sup> = −*k*<sup>2</sup> = *ik*, *i*

assuming *l*<sup>1</sup> = *l*<sup>2</sup> = *l* and *c*<sup>1</sup> = *c*<sup>2</sup> = *c*, Eq.(4) can be convert into the following form

where *F* is a polynomial of *u*(*x*, *y*, *t*) and its derivatives, *t* represents time variable and *x*, *y* represent spatial variables. Assume that there exists a transformation of unknown function

where *G* is a general polynomial in *Dt*, *Dx*, *Dy*, ··· , where the Hirota's bilinear operator

*<sup>∂</sup>x*� )*m*( *<sup>∂</sup>*

where *d* is a constant, *η*<sup>1</sup> = *k*1*x* + *l*1*y* + *c*1*t*, *η*<sup>2</sup> = *k*2*x* + *l*2*y* + *c*2*t*, and *kj*, *lj*, *cj*, *j* = 1, 2 are real

In order to get more forms solution of Eq.(1), we use a more general Ans*atz* ¨ which contains

*η*<sup>1</sup> = *k*1*x* + *l*1*y* + *c*1*t*

*<sup>∂</sup><sup>t</sup>* <sup>−</sup> *<sup>∂</sup> ∂t*

*F*(*u*, *ut*, *ux*, *uy*, ···) = 0 (1)

*G*(*Dt*, *Dx*, *Dy*, ···)*f* · *f* = 0 (2)

� )*n*[ *f*(*x*,*t*)*g*(*x*�

*f* = 1 + *eη*<sup>1</sup> + *eη*<sup>2</sup> + *deη*1+*η*<sup>2</sup> (4)

*f* = 1 + cos(*kx*)*ely*+*ct* + *de*2*ly*+2*ct* (5)

*f* = 1 + cos(*η*1)*eη*<sup>2</sup> + *de*2*η*<sup>2</sup> (6)

*<sup>η</sup>*<sup>2</sup> <sup>=</sup> *<sup>k</sup>*2*<sup>x</sup>* <sup>+</sup> *<sup>l</sup>*2*<sup>y</sup>* <sup>+</sup> *<sup>c</sup>*2*<sup>t</sup>* (7)

, *t* � )]| *x*� =*x*,*t* � <sup>=</sup>*<sup>t</sup>* (3)

<sup>2</sup> <sup>=</sup> <sup>−</sup>1 and

fusion of multi-wave.

**2.1. Homoclinic test approach**

D-operator is defined by[2]

*D<sup>m</sup> <sup>x</sup> D<sup>n</sup>*

where

$$f = e^{\eta\_2} \left[ \sqrt{d} \cosh(\eta\_2 + \ln(\sqrt{d})) + \cos(\eta\_1) \right] \tag{9}$$

We replace Eq.(9) with

$$f = \sqrt{d}\cosh(\eta\_2 + \ln(\sqrt{d})) + b\_1 \cos(\eta\_1) \tag{10}$$

where *b*1, *d* are constants. Now, factoring out the exponentials produces: Exploiting this Ans*atz* ¨ to obtain the new solutions of nonlinear evolution equation is called "Extended Homoclinic test approach". To derive analytic expression, we can take the following procedure in detail: inserting Eq.(10) into Eq.(2), then equating the coefficients of the same kind terms to zero, subsequently, solving the resulting algebraic equations to determine the relationship between variables *kj*, *lj*, *j* = 1, 2, ··· with the help of symbolic computation software such as Maple.

#### **2.3. Three-wave method**

Multi-wave solutions are important because they reveal the interactions between the inner-waves and the various frequency and velocity components. The whole multi-wave solution, for instance, may sometimes be converted into a single soliton of very high energy that propagates over large regions of space without dispersing and an extremely destructive wave is therefore produced of which the tsunami is a good example. Since all double-wave solutions can be found by using the exp-function method proposed by Fu and Dai [18], we propose an extension of the three-soliton method [6] in this section (called the three-wave method) for finding coupled wave solutions. Consider a high dimensional nonlinear evolution equation of the general form

$$F(\mathfrak{u}, \mathfrak{u}\_{\mathfrak{t}\prime} \mathfrak{u}\_{\mathfrak{X}\prime} \mathfrak{u}\_{\mathfrak{Y}\prime} \mathfrak{u}\_{\mathfrak{Z}\prime} \mathfrak{u}\_{\mathfrak{X}\prime} \cdots) = 0 \tag{11}$$

where *u* = *u*(*x*, *y*, *z*, *t*) and *F* is a polynomial *u* of and its derivatives, *t* represents time variable and *x*, *y*, *z* represent spatial variables. The three-wave method operates as follows. Step 1: By Painleve analysis, a transformation

$$
\mu = T(f) \tag{12}
$$

is made for some new and unknown function *f* . Step 2: Convert Eq. (11) into Hirota bilinear form:

$$H(D\_{l\prime}D\_{x\prime}D\_{y\prime}D\_{z\prime}\cdots)f \cdot f = 0\tag{13}$$

(i). Initial solution as a parameter is introduced to Eq.(20). We assume that the *u*<sup>0</sup> is an initial

to convert nonlinear evolution equation (20) into Hirota bilinear form which contains the small

where *D* is also the Hirota D-operator. The next step is to exploit the existent method to solving the Eq.(22). The perturbation parameter *u*<sup>0</sup> plays an important role to the resulting solution, where the spatiotemporal feature in multi-wave propagation including velocity and

(ii). The velocity of a wave variable as a parameter is introduced to Eq.(20). We can assume that *ξ* = *z* − *αt* in Eq.(20), where *ξ* is a new wave variable and *α* is its velocity. Then Eq.(20)

Then solving Eq.(23). In this case, the spatiotemporal feature in multi-wave propagation for

of remarkable non-linear problems both in physics and mathematics, the solutions of PKP equation have been studied extensively in various aspects. By applying various methods and techniques to PKP equation, exact traveling wave solutions, linearly solitary wave solutions, soliton-like solutions and some numerical solutions have been obtained[1-3,5-7]. Recently, two soliton and periodic soliton solutions were presented, resonance and non-resonance

In this section, we use homoclinic test method and extended homoclinic test technique to seek periodic soliton solution, exact periodic kink-wave solution, periodic soliton solution and doubly periodic solution of PKP equation. Furthermore, it is explicitly exhibited that the feature of solution is different varying with direction of wave propagation on the *x* -axis, periodic soliton is degenerated into doubly periodic wave when the direction of a wave

In this section, the periodic solition solution is constructed by homoclinic test technique and

Potential Kadomtsev-Petviashvili (PKP) equation studied in this section is described as

interactions between periodic soliton and different line solitons were investigated[6].

direction even the shape will change as *u*<sup>0</sup> makes small perturbation.

Eq.(20) may happen outstanding change as *α* makes a small perturbation.

**3. Potential Kadomtsev-Petviashvili equation**

propagation changes from progressing to the left into right.

bilinear form method and the degeneracy of solitary wave is investigated.

**3.1. Degenerative periodic solitary wave**

*u* = *u*<sup>0</sup> + *T*(*f*) (21)

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 109

*H*(*u*0, *Dt*, *Dx*, *Dy*, *Dz*, ···)*f* · *f* = 0 (22)

*F*1(*u*, *α*, *uξ*, *ux*, *uy*, *uxx*, ···) = 0 (23)

*uxt* + 6*uxuxx* + *uxxxx* + *uyy* = 0 (24)

*<sup>t</sup>* → *R*. It is well known that the PKP equation arose in number

solution of Eq.(20), then we use the transformation

perturbation parameters *u*<sup>0</sup> :

can be became to

where *<sup>u</sup>* : *Rx* <sup>×</sup> *Ry* <sup>×</sup> *<sup>R</sup>*<sup>+</sup>

where *D* is identical to the Eq.(3).

Step 3: Traditionally, we taking the following Ans*atz* ¨ to obtain the three soliton solution

$$f = 1 + e^{\tilde{\xi}\_1} + e^{\tilde{\xi}\_2} + e^{\tilde{\xi}\_3} + a\_{12}e^{\tilde{\xi}\_1 + \tilde{\xi}\_2} + a\_{13}e^{\tilde{\xi}\_1 + \tilde{\xi}\_3} + a\_{23}e^{\tilde{\xi}\_2 + \tilde{\xi}\_3} + a\_{123}e^{\tilde{\xi}\_1 + \tilde{\xi}\_2 + \tilde{\xi}\_3} \tag{14}$$

where

$$\mathfrak{F}\_{\mathfrak{j}} = a\_{\mathfrak{j}}\mathfrak{x} + b\_{\mathfrak{j}}\mathfrak{y} + c\_{\mathfrak{j}}\mathfrak{z} + d\_{\mathfrak{j}}t, \qquad \mathfrak{j} = 1, \mathfrak{Z}, \mathfrak{Z} \tag{15}$$

*a*13, *a*23, *a*<sup>123</sup> are the constants. Eq.(14) can be rewritten as

$$f = e^{\frac{\eta}{2}} \left( e^{-\eta\_1} + e^{\eta\_2} + e^{\eta\_3} + e^{\eta\_4} + a\_{12}e^{\eta\_4} + a\_{13}e^{\eta\_3} + a\_{23}e^{-\eta\_2} + a\_{123}e^{\eta\_1} \right) \tag{16}$$

where

$$\eta\_1 = \frac{\xi\_1 + \xi\_2 + \xi\_3}{2}, \qquad \eta\_2 = \frac{\xi\_1 - \xi\_2 - \xi\_3}{2}, \qquad \eta\_3 = \frac{-\xi\_1 + \xi\_2 - \xi\_3}{2}, \qquad \eta\_4 = \frac{-\xi\_1 - \xi\_2 + \xi\_3}{2} \tag{17}$$

Thus, this three soliton Ans*atz* ¨ contains four variables *η*1, *η*2, *η*<sup>3</sup> and *η*4. Here, we treat it in a different way. We factor out the *e η*1 <sup>2</sup> in Eq.(16) and decrease the numbers of variables to three terms. On the other hand, we set some parameters in a complex way. At last, the above analysis allows us to construct the following assumptions:

$$f = e^{-\xi\_1} + \delta\_1 \cos(\xi\_2) + \delta\_2 \cosh(\xi\_3) + \delta\_3 e^{\xi\_1} \tag{18}$$

or

$$f = e^{-\vec{\xi}\_1} + \delta\_1 \cos(\vec{\xi}\_2) + \delta\_2 \sinh(\vec{\xi}\_3) + \delta\_3 e^{\vec{\xi}\_1} \tag{19}$$

where *δj*, *j* = 1, 2, 3 are constants. In fact, from Eq.(18) or Eq.(19), it is easily seen that it only contains three wave variables. As a result, we call this method "three wave method". It is obvious that three-wave method is the extension and improvement of the traditional three-soliton method.

Step 4: Substitute Eq.(18) (or Eq.(19)) into Eq.(13), and collect the coefficients of *e*−*ξ*<sup>1</sup> , *eξ*<sup>1</sup> , sin(*ξ*2), cos(*ξ*2), cosh(*ξ*3) and sinh(*ξ*3). Then equate the coefficients of these terms to zero and obtain a set of over-determined algebraic equations in *aj*, *bj*, *cj* and *dj*, *j* = 1, 2, 3.

Step 5: Solve the set of algebraic equations in Step 4 using Maple and solve for *aj*, *bj*, *cj*, *dj* and *δj*, *j* = 1, 2, 3.

Step 6: Substituting the identified values into Eq.(12) and Eq.(13). Thus, we can deduce the exact multi-wave solutions of Eq.(11).

#### **2.4. Introducing parameters and small perturbation method**

In this section, we still consider a high dimensional nonlinear evolution equation of the general form

$$F(\mathfrak{u}\_{\prime}\mathfrak{u}\_{\prime\prime}\mathfrak{u}\_{\mathfrak{X}\prime}\mathfrak{u}\_{\mathcal{Y}\prime}\mathfrak{u}\_{\mathfrak{Z}\prime}\mathfrak{u}\_{\mathfrak{X}\prime}\cdots) = \mathbf{0} \tag{20}$$

where *u* = *u*(*x*, *y*, *z*, *t*) and *F* is a polynomial *u* of and its derivatives, *t* represents time variable and *x*, *y*, *z* represent spatial variables. Here, we introduce a parameter to Eq.(20) in two ways. (i). Initial solution as a parameter is introduced to Eq.(20). We assume that the *u*<sup>0</sup> is an initial solution of Eq.(20), then we use the transformation

$$
\mu = \mu\_0 + T(f) \tag{21}
$$

to convert nonlinear evolution equation (20) into Hirota bilinear form which contains the small perturbation parameters *u*<sup>0</sup> :

$$H(\mu\_0, D\_{\text{lt}}, D\_{\text{x}}, D\_{\text{y}}, D\_{\text{z}}, \dots)f \cdot f = 0 \tag{22}$$

where *D* is also the Hirota D-operator. The next step is to exploit the existent method to solving the Eq.(22). The perturbation parameter *u*<sup>0</sup> plays an important role to the resulting solution, where the spatiotemporal feature in multi-wave propagation including velocity and direction even the shape will change as *u*<sup>0</sup> makes small perturbation.

(ii). The velocity of a wave variable as a parameter is introduced to Eq.(20). We can assume that *ξ* = *z* − *αt* in Eq.(20), where *ξ* is a new wave variable and *α* is its velocity. Then Eq.(20) can be became to

$$F\_1(\mu, \mu\_\tau \mu\_{\tilde{\xi}\prime} \mu\_{\tilde{\omega}\prime} \mu\_{\tilde{\theta}\prime} \mu\_{\text{xx}\prime} \cdots) = 0 \tag{23}$$

Then solving Eq.(23). In this case, the spatiotemporal feature in multi-wave propagation for Eq.(20) may happen outstanding change as *α* makes a small perturbation.

#### **3. Potential Kadomtsev-Petviashvili equation**

4 Will-be-set-by-IN-TECH

Step 3: Traditionally, we taking the following Ans*atz* ¨ to obtain the three soliton solution

*f* = 1 + *eξ*<sup>1</sup> + *eξ*<sup>2</sup> + *eξ*<sup>3</sup> + *a*12*eξ*1+*ξ*<sup>2</sup> + *a*13*eξ*1+*ξ*<sup>3</sup> + *a*23*eξ*2+*ξ*<sup>3</sup> + *a*123*eξ*1+*ξ*2+*ξ*<sup>3</sup> (14)

<sup>2</sup> , *<sup>η</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>*ξ*1+*ξ*2−*ξ*<sup>3</sup>

Thus, this three soliton Ans*atz* ¨ contains four variables *η*1, *η*2, *η*<sup>3</sup> and *η*4. Here, we treat it in

three terms. On the other hand, we set some parameters in a complex way. At last, the above

where *δj*, *j* = 1, 2, 3 are constants. In fact, from Eq.(18) or Eq.(19), it is easily seen that it only contains three wave variables. As a result, we call this method "three wave method". It is obvious that three-wave method is the extension and improvement of the traditional

Step 4: Substitute Eq.(18) (or Eq.(19)) into Eq.(13), and collect the coefficients of *e*−*ξ*<sup>1</sup> , *eξ*<sup>1</sup> , sin(*ξ*2), cos(*ξ*2), cosh(*ξ*3) and sinh(*ξ*3). Then equate the coefficients of these terms to zero and

Step 5: Solve the set of algebraic equations in Step 4 using Maple and solve for *aj*, *bj*, *cj*, *dj* and

Step 6: Substituting the identified values into Eq.(12) and Eq.(13). Thus, we can deduce the

In this section, we still consider a high dimensional nonlinear evolution equation of the

where *u* = *u*(*x*, *y*, *z*, *t*) and *F* is a polynomial *u* of and its derivatives, *t* represents time variable and *x*, *y*, *z* represent spatial variables. Here, we introduce a parameter to Eq.(20) in two ways.

obtain a set of over-determined algebraic equations in *aj*, *bj*, *cj* and *dj*, *j* = 1, 2, 3.

**2.4. Introducing parameters and small perturbation method**

*η*1

*H*(*Dt*, *Dx*, *Dy*, *Dz*, ···)*f* · *f* = 0 (13)

*ξ<sup>j</sup>* = *ajx* + *bjy* + *cjz* + *djt*, *j* = 1, 2, 3 (15)

*f* = *e*−*ξ*<sup>1</sup> + *δ*<sup>1</sup> cos(*ξ*2) + *δ*<sup>2</sup> cosh(*ξ*3) + *δ*3*eξ*<sup>1</sup> (18)

*f* = *e*−*ξ*<sup>1</sup> + *δ*<sup>1</sup> cos(*ξ*2) + *δ*<sup>2</sup> sinh(*ξ*3) + *δ*3*eξ*<sup>1</sup> (19)

*F*(*u*, *ut*, *ux*, *uy*, *uz*, *uxx*, ···) = 0 (20)

<sup>2</sup> , *<sup>η</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>*ξ*1−*ξ*2+*ξ*<sup>3</sup>

<sup>2</sup> in Eq.(16) and decrease the numbers of variables to

<sup>2</sup> (17)

<sup>2</sup> (*e*−*η*<sup>1</sup> + *eη*<sup>2</sup> + *eη*<sup>3</sup> + *eη*<sup>4</sup> + *a*12*eη*<sup>4</sup> + *a*13*eη*<sup>3</sup> + *a*23*e*−*η*<sup>2</sup> + *a*123*eη*<sup>1</sup> ) (16)

is made for some new and unknown function *f* . Step 2: Convert Eq. (11) into Hirota bilinear form:

*a*13, *a*23, *a*<sup>123</sup> are the constants. Eq.(14) can be rewritten as

<sup>2</sup> , *<sup>η</sup>*<sup>2</sup> <sup>=</sup> *<sup>ξ</sup>*1−*ξ*2−*ξ*<sup>3</sup>

analysis allows us to construct the following assumptions:

where *D* is identical to the Eq.(3).

*f* = *e η*1

*<sup>η</sup>*<sup>1</sup> = *<sup>ξ</sup>*1+*ξ*2+*ξ*<sup>3</sup>

a different way. We factor out the *e*

exact multi-wave solutions of Eq.(11).

where

where

or

three-soliton method.

*δj*, *j* = 1, 2, 3.

general form

Potential Kadomtsev-Petviashvili (PKP) equation studied in this section is described as

$$
\mu\_{x\dagger} + \theta \mu\_x \mu\_{xx} + \mu\_{xxxx} + \mu\_{yy} = 0 \tag{24}
$$

where *<sup>u</sup>* : *Rx* <sup>×</sup> *Ry* <sup>×</sup> *<sup>R</sup>*<sup>+</sup> *<sup>t</sup>* → *R*. It is well known that the PKP equation arose in number of remarkable non-linear problems both in physics and mathematics, the solutions of PKP equation have been studied extensively in various aspects. By applying various methods and techniques to PKP equation, exact traveling wave solutions, linearly solitary wave solutions, soliton-like solutions and some numerical solutions have been obtained[1-3,5-7]. Recently, two soliton and periodic soliton solutions were presented, resonance and non-resonance interactions between periodic soliton and different line solitons were investigated[6].

In this section, we use homoclinic test method and extended homoclinic test technique to seek periodic soliton solution, exact periodic kink-wave solution, periodic soliton solution and doubly periodic solution of PKP equation. Furthermore, it is explicitly exhibited that the feature of solution is different varying with direction of wave propagation on the *x* -axis, periodic soliton is degenerated into doubly periodic wave when the direction of a wave propagation changes from progressing to the left into right.

#### **3.1. Degenerative periodic solitary wave**

In this section, the periodic solition solution is constructed by homoclinic test technique and bilinear form method and the degeneracy of solitary wave is investigated.

#### 6 Will-be-set-by-IN-TECH 110 Wave Processes in Classical and New Solids Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations <sup>7</sup>

Setting *ξ* = *x* − *αt* in Eq.(24) yields

$$
\mu\_{yy} - \alpha \mu\_{\tilde{\xi}\tilde{\xi}} + \theta \mu\_{\tilde{\xi}} \mu\_{\tilde{\xi}\tilde{\xi}} + \mu\_{\tilde{\xi}\tilde{\xi}\tilde{\xi}\tilde{\xi}} = 0 \tag{25}
$$

where *α* is a wave velocity. Let *u* = (ln *F*)*ξ*, then Eq.(25) can be reduced into the following bilinear form

$$\mathbb{E}\left[D\_y^2 - \alpha D\_\xi^2 + D\_\xi^4 - A\right]F \cdot F = 0\tag{26}$$


20

u



where parameters *b*3, *b*4, Ω1, *p*<sup>1</sup> satisfy following dispersive relations

Ω2 <sup>1</sup> <sup>=</sup> *<sup>p</sup>*<sup>4</sup>

In particularly, taking *γ*<sup>1</sup> = 0, *b*<sup>4</sup> = 1 in Eq.(35) yields

We can see that Ω<sup>2</sup>

as

where

0

*<sup>u</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>*b*<sup>3</sup> *<sup>p</sup>*<sup>1</sup> sin(*p*1*ξ*)*e*Ω1*η*+*γ*<sup>1</sup>

<sup>1</sup> *<sup>b</sup>*<sup>2</sup>

Taking *ξ* = *x* − *αt*, *η* = *iy* in Eq.(33), the exact solution to PKP equation (*α* ≥ 0) is expressed

*<sup>α</sup>* <sup>≥</sup> 0, *<sup>p</sup>*<sup>2</sup>

<sup>1</sup> ≥ 0

<sup>1</sup> <sup>+</sup> *<sup>α</sup>p*<sup>2</sup> 1

<sup>1</sup> ≥ 0 always holds for every real *p*<sup>1</sup> as long as *α* ≥ 0.

<sup>1</sup> <sup>+</sup> *<sup>α</sup>p*<sup>2</sup>

*<sup>u</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>*b*<sup>3</sup> *<sup>p</sup>* sin(*p*1(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*))

⎧ ⎪⎪⎪⎨

Ω<sup>2</sup> <sup>1</sup> <sup>=</sup> *<sup>p</sup>*<sup>4</sup>

*b*2 <sup>3</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup> <sup>1</sup> *b*<sup>4</sup> Ω<sup>2</sup> 1+3*p*<sup>4</sup> 1

*<sup>u</sup>* <sup>=</sup> <sup>−</sup> *<sup>b</sup>*<sup>3</sup> *<sup>p</sup>*<sup>1</sup> sin(*p*1(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*))

Obviously, both cos *p*1(*x* − *αt*) and cos(Ω1*y*) are periodic, so the solution given by Eq.(37) is an doubly periodic solution which is a periodic traveling wave progressing to the right with velocity *α* on the *x*-axis, and meanwhile is a periodic standing wave on the *y*-axis (see Fig.2).

and Eq.(37) from the constraint condition Eq.(31) and Eq.(36) respectively, where *k* > 4 is an

⎪⎪⎪⎩

It is important that we can take the same *p* and *p*<sup>1</sup> such that *p*<sup>2</sup> = *p*<sup>2</sup>

**Figure 1.** The periodic soliton solution which is a periodic wave progressing to the left with velocity

x

10

20


<sup>3</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup>

Ω<sup>2</sup> <sup>1</sup> <sup>+</sup> <sup>3</sup>*p*<sup>4</sup> 1

y

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 111

<sup>1</sup> <sup>+</sup> <sup>2</sup>*b*<sup>3</sup> cos(*p*1*ξ*)*e*Ω1*η*+*γ*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*4*e*2Ω1*η*+2*γ*<sup>1</sup> (33)

<sup>1</sup>*b*<sup>4</sup>

<sup>2</sup>*b*<sup>3</sup> cos(*p*1(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*)) + (*e*−*i*Ω1*y*−*γ*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*4*ei*Ω<sup>1</sup> *<sup>y</sup>*+*γ*<sup>1</sup> ) (35)

*<sup>b</sup>*<sup>3</sup> cos(*p*1(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*)) + cos (Ω1*y*) (37)

<sup>1</sup> <sup>=</sup> <sup>|</sup> *<sup>α</sup>*

*<sup>k</sup>* <sup>|</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup>

<sup>0</sup> in Eq.(30)

(34)

(36)

where *A* is an integration constant, *D<sup>m</sup> <sup>x</sup> D<sup>k</sup> <sup>t</sup>* is defined in Eq.(3). With regard to Eq.(26), using the homoclinic test technique, we can seek the solution in the form

$$F = 1 + b\_1(e^{ip\xi} + e^{-ip\xi})e^{\Omega y + \gamma} + b\_2e^{2\Omega y + 2\gamma} \tag{27}$$

where *A*, *p*, Ω, *γ*, *b*<sup>1</sup> and *b*<sup>2</sup> are all real to be determined below.

Substituting Eq.(27) into Eq.(26) yields the exact solution of Eq.(24) in the form

$$\mu = \frac{-2b\_1 p e^{\Omega y + \gamma} \sin(p\xi)}{1 + 2b\_1 \cos(p\xi) e^{\Omega y + \gamma} + b\_2 e^{2\Omega y + 2\gamma}}\tag{28}$$

where parameters *A*, *b*1, *b*2, Ω, *p* and *γ* satisfy dispersive relations

$$A = 0 \qquad \qquad \Omega^2 = -p^4 - ap^2 \qquad \qquad b\_1^2 = \frac{\Omega^2 b\_2}{\Omega^2 - 3p^4} \tag{29}$$

Obviously, *α* < 0 is required so that the conditions Ω<sup>2</sup> > 0, *b*<sup>2</sup> <sup>1</sup> <sup>&</sup>gt; 0 and 0 <sup>&</sup>lt; *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>*<sup>α</sup>* can be satisfied. Taking *ξ* = *x* − *αt*, *b*<sup>2</sup> = 1, *γ* = 0 in Eq.(28), then *b*<sup>1</sup> = � Ω<sup>2</sup> <sup>Ω</sup><sup>2</sup>−3*p*<sup>4</sup> <sup>&</sup>gt; 1, the exact solution to PKP equation can be expressed as

$$u = \frac{-b\_1 p \sin(p(\mathbf{x} - \alpha t))}{b\_1 \cos(p(\mathbf{x} - \alpha t)) + \cosh(\Omega y)}\tag{30}$$

where

$$\begin{cases} a < 0\\ 0 < p^2 < -\frac{a}{4} \\ \Omega^2 = -p^4 - ap^2 \\ b\_1^2 = \frac{\Omega^2}{\Omega^2 - 3p^4} \end{cases} \tag{31}$$

Here, we choice 0 <sup>&</sup>lt; *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup> *<sup>α</sup>* <sup>4</sup> such that *<sup>b</sup>*<sup>2</sup> <sup>1</sup> > 0, then Eq.(30) is the periodic soliton solution of PKP equation which is a periodic traveling wave progressing to the left with velocity |*α*| on the *x*-axis, and meanwhile is a soliton on the *y*-axis (see Fig.1).

Making a variable transformation *ξ* = *x* − *αt*, *η* = *iy* in Eq.(24), then it can be transformed into the following form

$$
\mu\_{\eta\eta} - (-\mathfrak{a})\mu\_{\tilde{\xi}\tilde{\xi}} - 6\mu\_{\tilde{\xi}}\mu\_{\tilde{\xi}\tilde{\xi}} - \mu\_{\tilde{\xi}\tilde{\xi}\tilde{\xi}\tilde{\xi}} = 0 \tag{32}
$$

Letting *u* = (ln *F*)*ξ*, being similar to the way of dealing with PKP equation in above, we take

$$F = 1 + b\_{\mathfrak{Z}} (e^{ip\_1 \mathfrak{Z}} + e^{-ip\_1 \mathfrak{Z}}) e^{\Omega\_1 \eta + \gamma\_1} + b\_{\mathfrak{A}} e^{2 \Omega\_1 \eta + 2 \gamma\_1}$$

110 Wave Processes in Classical and New Solids Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations <sup>7</sup> Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 111

**Figure 1.** The periodic soliton solution which is a periodic wave progressing to the left with velocity |*α*| > 0 on the *x*-axis ,and meanwhile is a soliton on the *y*-axis

By computing, the exact solution of Eq.(32) is given by

$$\mu = -\frac{2b\_3 p\_1 \sin(p\_1 \xi) e^{\Omega\_1 \eta + \gamma\_1}}{1 + 2b\_3 \cos(p\_1 \xi) e^{\Omega\_1 \eta + \gamma\_1} + b\_4 e^{2\Omega\_1 \eta + 2\gamma\_1}}\tag{33}$$

where parameters *b*3, *b*4, Ω1, *p*<sup>1</sup> satisfy following dispersive relations

$$
\Omega\_1^2 = p\_1^4 + ap\_1^2 \qquad \qquad b\_3^2 = \frac{\Omega\_1^2 b\_4}{\Omega\_1^2 + 3p\_1^4} \tag{34}
$$

We can see that Ω<sup>2</sup> <sup>1</sup> ≥ 0 always holds for every real *p*<sup>1</sup> as long as *α* ≥ 0.

Taking *ξ* = *x* − *αt*, *η* = *iy* in Eq.(33), the exact solution to PKP equation (*α* ≥ 0) is expressed as

$$u = -\frac{2b\_3 p \sin(p\_1(\mathbf{x} - at))}{2b\_3 \cos(p\_1(\mathbf{x} - at)) + (e^{-i\Omega\_1 y - \gamma\_1} + b\_4 e^{i\Omega\_1 y + \gamma\_1})}\tag{35}$$

where

6 Will-be-set-by-IN-TECH

where *α* is a wave velocity. Let *u* = (ln *F*)*ξ*, then Eq.(25) can be reduced into the following

<sup>−</sup>*ip<sup>ξ</sup>* )*e*

<sup>Ω</sup>*y*+*<sup>γ</sup>* + *b*2*e*

*<sup>ξ</sup>* <sup>+</sup> *<sup>D</sup>*<sup>4</sup>

*<sup>x</sup> D<sup>k</sup>*

*ip<sup>ξ</sup>* + *e*

Substituting Eq.(27) into Eq.(26) yields the exact solution of Eq.(24) in the form

*<sup>A</sup>* <sup>=</sup> <sup>0</sup> <sup>Ω</sup><sup>2</sup> <sup>=</sup> <sup>−</sup>*p*<sup>4</sup> <sup>−</sup> *<sup>α</sup>p*<sup>2</sup> *<sup>b</sup>*<sup>2</sup>

*<sup>u</sup>* <sup>=</sup> <sup>−</sup>*b*1*<sup>p</sup>* sin(*p*(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*))

<sup>0</sup> <sup>&</sup>lt; *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup> *<sup>α</sup>*

PKP equation which is a periodic traveling wave progressing to the left with velocity |*α*| on

Making a variable transformation *ξ* = *x* − *αt*, *η* = *iy* in Eq.(24), then it can be transformed into

Letting *u* = (ln *F*)*ξ*, being similar to the way of dealing with PKP equation in above, we take

<sup>−</sup>*ip*1*<sup>ξ</sup>* )*e*

4 <sup>Ω</sup><sup>2</sup> <sup>=</sup> <sup>−</sup>*p*<sup>4</sup> <sup>−</sup> *<sup>α</sup>p*<sup>2</sup>

*α* < 0

*<sup>u</sup>* <sup>=</sup> <sup>−</sup>2*b*<sup>1</sup> *pe*Ω*y*+*<sup>γ</sup>* sin(*pξ*)

[*D*<sup>2</sup> *<sup>y</sup>* <sup>−</sup> *<sup>α</sup>D*<sup>2</sup>

the homoclinic test technique, we can seek the solution in the form

*F* = 1 + *b*1(*e*

where *A*, *p*, Ω, *γ*, *b*<sup>1</sup> and *b*<sup>2</sup> are all real to be determined below.

where parameters *A*, *b*1, *b*2, Ω, *p* and *γ* satisfy dispersive relations

Obviously, *α* < 0 is required so that the conditions Ω<sup>2</sup> > 0, *b*<sup>2</sup>

solution to PKP equation can be expressed as

Here, we choice 0 <sup>&</sup>lt; *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup> *<sup>α</sup>*

the following form

be satisfied. Taking *ξ* = *x* − *αt*, *b*<sup>2</sup> = 1, *γ* = 0 in Eq.(28), then *b*<sup>1</sup> =

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

<sup>4</sup> such that *<sup>b</sup>*<sup>2</sup>

the *x*-axis, and meanwhile is a soliton on the *y*-axis (see Fig.1).

*F* = 1 + *b*3(*eip*1*<sup>ξ</sup>* + *e*

*b*2 <sup>1</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup> Ω<sup>2</sup>−3*p*<sup>4</sup>

*uyy* − *αuξξ* + 6*uξuξξ* + *uξξξξ* = 0 (25)

<sup>1</sup> <sup>+</sup> <sup>2</sup>*b*<sup>1</sup> cos(*pξ*)*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>b</sup>*2*e*2Ω*y*+2*<sup>γ</sup>* (28)

<sup>1</sup> <sup>=</sup> <sup>Ω</sup>2*b*<sup>2</sup>

*<sup>b</sup>*<sup>1</sup> cos(*p*(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*)) + cosh(Ω*y*) (30)

<sup>1</sup> > 0, then Eq.(30) is the periodic soliton solution of

2Ω1*η*+2*γ*<sup>1</sup>

*uηη* − (−*α*)*uξξ* − 6*uξuξξ* − *uξξξξ* = 0 (32)

<sup>Ω</sup>1*η*+*γ*<sup>1</sup> + *b*4*e*

*<sup>ξ</sup>* − *A*]*F* · *F* = 0 (26)

<sup>2</sup>Ω*y*+2*<sup>γ</sup>* (27)

<sup>Ω</sup><sup>2</sup> <sup>−</sup> <sup>3</sup>*p*<sup>4</sup> (29)

<sup>1</sup> <sup>&</sup>gt; 0 and 0 <sup>&</sup>lt; *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>*<sup>α</sup>* can

<sup>Ω</sup><sup>2</sup>−3*p*<sup>4</sup> <sup>&</sup>gt; 1, the exact

(31)

� Ω<sup>2</sup>

*<sup>t</sup>* is defined in Eq.(3). With regard to Eq.(26), using

Setting *ξ* = *x* − *αt* in Eq.(24) yields

where *A* is an integration constant, *D<sup>m</sup>*

bilinear form

where

$$\begin{cases} a \ge 0, \quad p\_1^2 \ge 0 \\ \Omega\_1^2 = p\_1^4 + a p\_1^2 \\ b\_3^2 = \frac{\Omega\_1^2 b\_4}{\Omega\_1^2 + 3 p\_1^4} \end{cases} \tag{36}$$

In particularly, taking *γ*<sup>1</sup> = 0, *b*<sup>4</sup> = 1 in Eq.(35) yields

$$u = -\frac{b\_3 p\_1 \sin(p\_1(\mathbf{x} - \mathbf{a}t))}{b\_3 \cos(p\_1(\mathbf{x} - \mathbf{a}t)) + \cos(\Omega\_1 y)}\tag{37}$$

Obviously, both cos *p*1(*x* − *αt*) and cos(Ω1*y*) are periodic, so the solution given by Eq.(37) is an doubly periodic solution which is a periodic traveling wave progressing to the right with velocity *α* on the *x*-axis, and meanwhile is a periodic standing wave on the *y*-axis (see Fig.2).

It is important that we can take the same *p* and *p*<sup>1</sup> such that *p*<sup>2</sup> = *p*<sup>2</sup> <sup>1</sup> <sup>=</sup> <sup>|</sup> *<sup>α</sup> <sup>k</sup>* <sup>|</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>0</sup> in Eq.(30) and Eq.(37) from the constraint condition Eq.(31) and Eq.(36) respectively, where *k* > 4 is an

**Figure 2.** The doubly periodic solution which is a periodic wave progressing to the right with velocity *α* on the *x*-axis, and meanwhile is a periodic standing wave on the *y*-axis.

arbitrary real number. Therefore, we obtain a periodic soliton solution and an doubly periodic solution which have the same period with *x*-direction as follows

$$\begin{cases} u = \frac{-b\_1 p\_0 \sin(p\_0(\mathbf{x} - at))}{b\_1 \cos(p\_0(\mathbf{x} - at)) + \cosh(\Omega y)}, \quad a < 0 \\\\ u = -\frac{b\_3 p\_0 \sin(p\_0(\mathbf{x} - at))}{b\_3 \cos(p\_0(\mathbf{x} - at)) + \cos(\Omega\_1 y)}, \quad a \ge 0 \end{cases} \tag{38}$$


u

*<sup>e</sup>p*(*x*+11*p*2*t*) <sup>−</sup> <sup>2</sup>

*ep*(*x*+11*p*2*t*) + 2

**Figure 3.** The periodic kink-wave solution

*b*1 2 √5

**4. Kadomtsev-Petviashvili equation**

coefficient is defined as follows: *s* = ±*i*, *i*

with the operator *∂*−<sup>1</sup> *<sup>x</sup>* defined by

equation, which can be written in normalized form as follows[4]:

*ut* <sup>−</sup> <sup>3</sup>(*u*2)*<sup>x</sup>* <sup>−</sup> *uxxx* <sup>+</sup> *<sup>s</sup>*

*<sup>u</sup>* <sup>=</sup> *<sup>p</sup>*[ *<sup>b</sup>*<sup>1</sup> 2 √5

It can be rewritten as

Specially, taking *b*<sup>1</sup> = 2

0

<sup>2</sup> <sup>x</sup> -2

<sup>√</sup>5 sin(*p*(*<sup>x</sup>* <sup>−</sup> <sup>11</sup>*p*2*<sup>t</sup>* <sup>±</sup> <sup>3</sup>*py*)) <sup>−</sup> <sup>2</sup>

<sup>√</sup>5 cos(*p*(*<sup>x</sup>* <sup>−</sup> <sup>11</sup>*p*2*<sup>t</sup>* <sup>±</sup> <sup>3</sup>*py*)) + <sup>2</sup>

*<sup>u</sup>*(*x*, *<sup>y</sup>*, *<sup>t</sup>*) = *<sup>p</sup>*[2 sinh(*p*(*<sup>x</sup>* <sup>+</sup> <sup>11</sup>*p*2*t*) + *<sup>γ</sup>*) <sup>−</sup> sin(*p*(*<sup>x</sup>* <sup>±</sup> <sup>3</sup>*py* <sup>−</sup> <sup>11</sup>*p*2*t*))]

It shows a periodic kink-wave whose speed is 11*p*<sup>2</sup> and period is 2*π*/3*p* of space variable *y*. It exhibits elastic interaction between a solitary wave and periodic wave with the same speed in opposed direction each other. It is an interesting phenomenon in fluid mechanics (see Fig.3).

The (2+1)-dimensional (two spatial and one temporal) generalization of Korteweg-de Vries equation (KdV) was given by Kadomtsev and Petviashvili to discuss the stability of (1+1)-dimensional soliton to the transverse long-wave disturbances, which is known as KP equation or the (2+1)-dimensional KdV equation. There are two distinct versions of KP

<sup>2</sup>*∂*−<sup>1</sup>

*<sup>x</sup> <sup>f</sup>*(*x*) = *<sup>x</sup>*

−∞

The propagation property of solitons depends essentially on the sign of *s*<sup>2</sup> in equation. The

for positive dispersion. Here *u* = *u*(*x*, *y*, *t*) is a scalar function, *x* and *y* are respectively the longitudinal and transverse spatial coordinates, the subscripts *x*, *y*, *t* denote partial derivatives. When *s* = ±*i*, it is usually called KPI, while for *s* = ±1, it is called KPII. KP equation is the natural generalization of the well known KdV equation (*ut* − 6*uux* − *uxxx* = 0)

*f*(*ξ*)*dξ*

*∂*−<sup>1</sup>

<sup>√</sup>5*<sup>k</sup>* and *<sup>γ</sup>* <sup>=</sup> ln *<sup>k</sup>*, *<sup>k</sup>* <sup>&</sup>gt; 1, yields a periodic kink solution as follows

2 cosh(*p*(*<sup>x</sup>* <sup>+</sup> <sup>11</sup>*p*2*t*) + *<sup>γ</sup>*) + cos(*p*(*<sup>x</sup>* <sup>±</sup> <sup>3</sup>*py* <sup>−</sup> <sup>11</sup>*p*2*t*)) (44)

*<sup>x</sup> uyy* = 0 (∗)

<sup>2</sup> <sup>=</sup> <sup>−</sup>1 for negative dispersion and *<sup>s</sup>* <sup>=</sup> <sup>±</sup><sup>1</sup>

0

2

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 113

y

√5

√5

*<sup>b</sup>*<sup>1</sup> *<sup>e</sup>*−*p*(*x*+11*p*2*t*)]

*<sup>b</sup>*<sup>1</sup> *<sup>e</sup>*−*p*(*x*+11*p*2*t*) (43)

It is easy to find that the feature of solution of Eq.(24) is different from Eq.(38) on the arbitrary small neighborhood of velocity *α* = 0. There exists a periodic soliton solution which is a periodic with *x*-direction, and a soliton with *y*-direction as well as *α* < 0. When *α* ≥ 0, this periodic soliton degenerates into a doubly periodic wave which are periodic with both *x*-direction and *y*-direction. However, period with *x*-direction is preserved identically.

#### **3.2. Exact periodic kink-wave solution**

In this section, a new type of periodic kink-wave solution for PKP equation is obtained using extended homoclinic test technique.

Taking a transformation

$$
\mu = (\ln F)\_{\mathfrak{x}} \tag{39}
$$

Then Eq.(24) can be transformed as

$$[D\_y^2 + D\_x D\_t + D\_x^4]F \cdot F = 0\tag{40}$$

In this case, we let

$$F = e^{-\Omega(\mathbf{x} + at) - \beta y} + b\_1 \cos(p(\mathbf{x} - at) + \beta y) + b\_2 e^{\Omega(\mathbf{x} + at)} \tag{41}$$

Following the procedure of extended homoclinic test technique, we derive the periodic kink-wave solution of PKP equation(see Fig.3)

$$u = \frac{p[-e^{-p(\mathbf{x} + 11p^2t)} - b\_1 \sin(p(\mathbf{x} - 11p^2t \pm 3py)) + \frac{b\_1^2}{20}e^{p(\mathbf{x} + 11p^2t)}]}{e^{-p(\mathbf{x} + 11p^2t)} + b\_1 \cos(p(\mathbf{x} - 11p^2t \pm 3py)) + \frac{b\_1^2}{20}e^{p(\mathbf{x} + 11p^2t)}} \tag{42}$$

**Figure 3.** The periodic kink-wave solution

It can be rewritten as

8 Will-be-set-by-IN-TECH


on the *x*-axis, and meanwhile is a periodic standing wave on the *y*-axis.

solution which have the same period with *x*-direction as follows

[*D*<sup>2</sup>

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

**3.2. Exact periodic kink-wave solution**

kink-wave solution of PKP equation(see Fig.3)

extended homoclinic test technique.

Then Eq.(24) can be transformed as

Taking a transformation

In this case, we let

u

0 <sup>5</sup> <sup>x</sup>

*<sup>u</sup>* <sup>=</sup> <sup>−</sup>*b*<sup>1</sup> *<sup>p</sup>*<sup>0</sup> sin(*p*0(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*))

*<sup>u</sup>* <sup>=</sup> <sup>−</sup> *<sup>b</sup>*<sup>3</sup> *<sup>p</sup>*<sup>0</sup> sin(*p*0(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*))

**Figure 2.** The doubly periodic solution which is a periodic wave progressing to the right with velocity *α*

arbitrary real number. Therefore, we obtain a periodic soliton solution and an doubly periodic

*b*<sup>1</sup> cos(*p*0(*x* − *αt*)) + cosh(Ω*y*)

*b*<sup>3</sup> cos(*p*0(*x* − *αt*)) + cos (Ω1*y*)

It is easy to find that the feature of solution of Eq.(24) is different from Eq.(38) on the arbitrary small neighborhood of velocity *α* = 0. There exists a periodic soliton solution which is a periodic with *x*-direction, and a soliton with *y*-direction as well as *α* < 0. When *α* ≥ 0, this periodic soliton degenerates into a doubly periodic wave which are periodic with both *x*-direction and *y*-direction. However, period with *x*-direction is preserved identically.

In this section, a new type of periodic kink-wave solution for PKP equation is obtained using

*<sup>y</sup>* + *DxDt* + *<sup>D</sup>*<sup>4</sup>

*<sup>u</sup>* <sup>=</sup> *<sup>p</sup>*[−*e*−*p*(*x*+11*p*2*t*) <sup>−</sup> *<sup>b</sup>*<sup>1</sup> sin(*p*(*<sup>x</sup>* <sup>−</sup> <sup>11</sup>*p*2*<sup>t</sup>* <sup>±</sup> <sup>3</sup>*py*)) + *<sup>b</sup>*<sup>2</sup>

*<sup>e</sup>*−*p*(*x*+11*p*2*t*) <sup>+</sup> *<sup>b</sup>*<sup>1</sup> cos(*p*(*<sup>x</sup>* <sup>−</sup> <sup>11</sup>*p*2*<sup>t</sup>* <sup>±</sup> <sup>3</sup>*py*)) + *<sup>b</sup>*<sup>2</sup>

Following the procedure of extended homoclinic test technique, we derive the periodic



y

, *α* < 0

, *α* ≥ 0

*u* = (ln *F*)*<sup>x</sup>* (39)

1

1

<sup>20</sup> *<sup>e</sup>p*(*x*+11*p*2*t*)]

<sup>20</sup> *<sup>e</sup>p*(*x*+11*p*2*t*) (42)

*<sup>F</sup>* <sup>=</sup> *<sup>e</sup>*−Ω(*x*+*αt*)−*β<sup>y</sup>* <sup>+</sup> *<sup>b</sup>*<sup>1</sup> cos(*p*(*<sup>x</sup>* <sup>−</sup> *<sup>α</sup>t*) + *<sup>β</sup>y*) + *<sup>b</sup>*2*e*Ω(*x*+*αt*) (41)

*<sup>x</sup>*]*F* · *F* = 0 (40)

(38)

$$u = \frac{p[\frac{b\_1}{2\sqrt{5}}e^{p(\mathbf{x} + 11p^2t)} - 2\sqrt{5}\sin(p(\mathbf{x} - 11p^2t \pm 3py)) - \frac{2\sqrt{5}}{b\_1}e^{-p(\mathbf{x} + 11p^2t)}]}{\frac{b\_1}{2\sqrt{5}}e^{p(\mathbf{x} + 11p^2t)} + 2\sqrt{5}\cos(p(\mathbf{x} - 11p^2t \pm 3py)) + \frac{2\sqrt{5}}{b\_1}e^{-p(\mathbf{x} + 11p^2t)}} \tag{43}$$

Specially, taking *b*<sup>1</sup> = 2 <sup>√</sup>5*<sup>k</sup>* and *<sup>γ</sup>* <sup>=</sup> ln *<sup>k</sup>*, *<sup>k</sup>* <sup>&</sup>gt; 1, yields a periodic kink solution as follows

$$u(\mathbf{x}, y, t) = \frac{p[2\sinh(p(\mathbf{x} + 11p^2t) + \gamma) - \sin(p(\mathbf{x} \pm 3py - 11p^2t))]}{2\cosh(p(\mathbf{x} + 11p^2t) + \gamma) + \cos(p(\mathbf{x} \pm 3py - 11p^2t))}\tag{44}$$

It shows a periodic kink-wave whose speed is 11*p*<sup>2</sup> and period is 2*π*/3*p* of space variable *y*. It exhibits elastic interaction between a solitary wave and periodic wave with the same speed in opposed direction each other. It is an interesting phenomenon in fluid mechanics (see Fig.3).

#### **4. Kadomtsev-Petviashvili equation**

The (2+1)-dimensional (two spatial and one temporal) generalization of Korteweg-de Vries equation (KdV) was given by Kadomtsev and Petviashvili to discuss the stability of (1+1)-dimensional soliton to the transverse long-wave disturbances, which is known as KP equation or the (2+1)-dimensional KdV equation. There are two distinct versions of KP equation, which can be written in normalized form as follows[4]:

$$
\mu\_l - \Im(u^2)\_{\ge} - \mu\_{\text{xxx}} + s^2 \partial\_{\ge}^{-1} \mu\_{\text{yy}} = 0 \tag{\*}
$$

with the operator *∂*−<sup>1</sup> *<sup>x</sup>* defined by

$$
\partial\_{\mathfrak{x}}^{-1}f(\mathfrak{x}) = \int\_{-\infty}^{\mathfrak{x}} f(\mathfrak{f})d\mathfrak{f}
$$

The propagation property of solitons depends essentially on the sign of *s*<sup>2</sup> in equation. The coefficient is defined as follows: *s* = ±*i*, *i* <sup>2</sup> <sup>=</sup> <sup>−</sup>1 for negative dispersion and *<sup>s</sup>* <sup>=</sup> <sup>±</sup><sup>1</sup> for positive dispersion. Here *u* = *u*(*x*, *y*, *t*) is a scalar function, *x* and *y* are respectively the longitudinal and transverse spatial coordinates, the subscripts *x*, *y*, *t* denote partial derivatives. When *s* = ±*i*, it is usually called KPI, while for *s* = ±1, it is called KPII. KP equation is the natural generalization of the well known KdV equation (*ut* − 6*uux* − *uxxx* = 0)

#### 10 Will-be-set-by-IN-TECH 114 Wave Processes in Classical and New Solids Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations <sup>11</sup>

from one to two spatial dimensions. It arises naturally in many other applications, particularly in plasma physics, gas dynamics, and elsewhere. Both KPI and KPII are exactly integrable via the Inverse Scattering Transformation. Kadomtsev and Petviashvili have shown that the line soliton of the KP equation is unstable in the case of positive dispersion and is stable for the negative dispersion[3,22]. The solutions of KP equation have been studied extensively in various aspects. The inclined periodic soliton solution and the lattice soliton solution were expressed as exact imbricate series of rational soliton solutions to KP equation with positive dispersion[22]. Resonant interaction of two-soliton among three obliquely oriented solitons in higher dimension was first studied by Miles[26]. Y.Kodama and Dai have proved that the KP equation provides line solitons in shallow water and these solitons can be of resonance[7, 27].

In this section, spatial-temporal bifurcation for KP equation is considered, several types of exact solutions to KP equation were constructed by bilinear form and homoclinic test approach. It is explicitly analyzed that the feature of the solitary wave is different on the both sides of equilibrium solution *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>6</sup> , which is a unique periodic bifurcation point for KPI and deflexion point of soliton for KPII. As for KPI, when the equilibrium *u*<sup>0</sup> varies from one side of <sup>−</sup><sup>1</sup> <sup>6</sup> to another side, two-solitary wave changes into doubly periodic wave. Whereas, the *y*-periodic solitary wave changes into *x* − *t*-periodic solitary wave for KPII, the propagation direction of periodic solitary wave occurs outstanding deflexion. In addition, some new type of multi-wave solutions are obtained using three-wave method.

#### **4.1. Spatiotemporal bifurcation and deflexion of the soliton**

Kadomtsev-Petviashvili (KP) equation in normalized variable *u*(*x*, *y*, *t*) reads

$$
\mu\_{\rm xt} - \mu\_{\rm xxxx} - \Im(\mu^2)\_{\rm xx} - s^2 \mu\_{\rm yy} = 0 \tag{45}
$$

where parameters satisfy

It is obviously that *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>−</sup><sup>1</sup>

where

can be satisfied and *u*<sup>0</sup> is a free parameter.

*u*1(*x*, *y*, *t*) = *u*<sup>0</sup> +

*<sup>A</sup>* <sup>=</sup> <sup>0</sup> <sup>Ω</sup><sup>2</sup> <sup>=</sup> <sup>−</sup>*p*<sup>4</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0)*p*<sup>2</sup> *<sup>b</sup>*<sup>2</sup>

2*p*2[4*b*<sup>2</sup>

⎧ ⎪⎪⎪⎪⎪⎪⎨ *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>−</sup><sup>1</sup> 6 *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>1+6*u*<sup>0</sup> 4 <sup>Ω</sup><sup>2</sup> <sup>=</sup> <sup>−</sup>*p*<sup>4</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0)*p*<sup>2</sup>

<sup>1</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup> *<sup>b</sup>*<sup>2</sup> Ω<sup>2</sup>−3*p*<sup>4</sup>

> 0 20

Letting *u* = *u*<sup>0</sup> + 2(ln *F*)*ξξ* and using a similar way dealing with KPI, we take

*F* = 1 + *b*1(*eip<sup>ξ</sup>* + *e*

40 x -40

Making a variable transformation *ξ* = *x* − *t* in Eq.(54), it can be transformed into the following

<sup>−</sup>*ip<sup>ξ</sup>* )*e*

4

<sup>Ω</sup>*y*+*<sup>γ</sup>* + *b*2*e*

<sup>1</sup> <sup>+</sup> *<sup>b</sup>*1(*eip<sup>ξ</sup>* <sup>+</sup> *<sup>e</sup>*−*ip<sup>ξ</sup>* )(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]

y

*uxt* <sup>−</sup> *uxxxx* <sup>−</sup> <sup>3</sup>(*u*2)*xx* <sup>+</sup> *uyy* <sup>=</sup> <sup>0</sup> (54)

*uyy* <sup>−</sup> *<sup>u</sup>ξξ* <sup>−</sup> <sup>3</sup>(*u*2)*ξξ* <sup>−</sup> *<sup>u</sup>ξξξξ* <sup>=</sup> <sup>0</sup> (55)

[*b*1(*eip<sup>ξ</sup>* <sup>+</sup> *<sup>e</sup>*−*ip<sup>ξ</sup>* )+(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]<sup>2</sup> (56)

2Ω*y*+2*γ*

⎪⎪⎪⎪⎪⎪⎩


u

**Figure 4.** The two-soliton solution for KPI equation as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

By computing, the exact solution of Eq.(55) is given by

*<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*<sup>0</sup> <sup>−</sup> <sup>2</sup>*p*2[4*b*<sup>2</sup>

KPII equation is given by

form

*b*2

<sup>6</sup> is required so that the conditions <sup>Ω</sup><sup>2</sup> <sup>&</sup>gt; 0, *<sup>b</sup>*<sup>2</sup>

In Eq.(50), taking *ξ* = *i*(*x* − *t*), we obtain the two-soliton solution of KPI equation (see Fig.4)

<sup>1</sup> <sup>+</sup> *<sup>b</sup>*1(*ep*(*x*−*t*) <sup>+</sup> *<sup>e</sup>*−*p*(*x*−*t*)

<sup>1</sup> <sup>=</sup> <sup>Ω</sup>2*b*<sup>2</sup>

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 115

)(*b*2*e*Ω*y*+*<sup>γ</sup>* + *e*−Ω*y*−*γ*)]

[*b*1(*ep*(*x*−*t*) <sup>+</sup> *<sup>e</sup>*−*p*(*x*−*<sup>t</sup>*))+(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]<sup>2</sup> (52)

<sup>Ω</sup><sup>2</sup> <sup>−</sup> <sup>3</sup>*p*<sup>4</sup> (51)

<sup>1</sup> <sup>&</sup>gt; 0 and *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>1+6*u*<sup>0</sup>

4

(53)

where *u* : *Rx* × *Ry* × *Rt* → *R*. It is easily to note that *u* = *u*<sup>0</sup> is an equilibrium solution of KP equation, where *u*<sup>0</sup> is an arbitrary constant.

Now, we consider KPI equation

$$
\mu\_{\rm xt} - \mu\_{\rm xxxx} - \Im(\mu^2)\_{\rm xx} - \mu\_{\rm yy} = 0 \tag{46}
$$

Setting *ξ* = *i*(*x* − *t*) gives

$$
u\_{yy} - 
u\_{\tilde{\xi}\tilde{\xi}} - \Im(\mu^2)\_{\tilde{\xi}\tilde{\xi}} + 
u\_{\tilde{\xi}\tilde{\xi}\tilde{\xi}\tilde{\xi}} = 0\tag{47}$$

Let *u* = *u*<sup>0</sup> − 2(ln *F*)*ξξ*, Eq.(46) can be transformed into the following bilinear form

$$\left[D\_y^2 - (1 + 6\mu\_0)D\_\xi^2 + D\_\xi^4 - A\right]F \cdot F = 0\tag{48}$$

Using "homoclinic test technique", we are going to seek the solution of the form

$$F = 1 + b\_1(e^{ip\tilde{\xi}} + e^{-ip\tilde{\xi}})e^{\Omega y + \gamma} + b\_2e^{2\Omega y + 2\gamma} \tag{49}$$

where *A*, *p*, Ω, *γ*, *b*<sup>1</sup> and *b*<sup>2</sup> are all real.

Substituting Eq.(49) into Eq.(48) with Eq.(47) yields the exact solution of Eq.(47) of the form

$$u = u\_0 + \frac{2p^2[4b\_1^2 + b\_1(e^{ip\xi} + e^{-ip\xi})(b\_2e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]}{[b\_1(e^{ip\xi} + e^{-ip\xi}) + (b\_2e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]^2} \tag{50}$$

where parameters satisfy

$$A = 0 \qquad \qquad \Omega^2 = -p^4 - (1 + 6\mu\_0)p^2 \qquad \qquad b\_1^2 = \frac{\Omega^2 b\_2}{\Omega^2 - 3p^4} \tag{51}$$

It is obviously that *<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>−</sup><sup>1</sup> <sup>6</sup> is required so that the conditions <sup>Ω</sup><sup>2</sup> <sup>&</sup>gt; 0, *<sup>b</sup>*<sup>2</sup> <sup>1</sup> <sup>&</sup>gt; 0 and *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>1+6*u*<sup>0</sup> 4 can be satisfied and *u*<sup>0</sup> is a free parameter.

In Eq.(50), taking *ξ* = *i*(*x* − *t*), we obtain the two-soliton solution of KPI equation (see Fig.4)

$$u\_1(\mathbf{x}, y, t) = u\_0 + \frac{2p^2[4b\_1^2 + b\_1(e^{p(\mathbf{x}-t)} + e^{-p(\mathbf{x}-t)})(b\_2e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]}{[b\_1(e^{p(\mathbf{x}-t)} + e^{-p(\mathbf{x}-t)}) + (b\_2e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]^2} \tag{52}$$

where

10 Will-be-set-by-IN-TECH

from one to two spatial dimensions. It arises naturally in many other applications, particularly in plasma physics, gas dynamics, and elsewhere. Both KPI and KPII are exactly integrable via the Inverse Scattering Transformation. Kadomtsev and Petviashvili have shown that the line soliton of the KP equation is unstable in the case of positive dispersion and is stable for the negative dispersion[3,22]. The solutions of KP equation have been studied extensively in various aspects. The inclined periodic soliton solution and the lattice soliton solution were expressed as exact imbricate series of rational soliton solutions to KP equation with positive dispersion[22]. Resonant interaction of two-soliton among three obliquely oriented solitons in higher dimension was first studied by Miles[26]. Y.Kodama and Dai have proved that the KP equation provides line solitons in shallow water and these solitons can be of resonance[7,

In this section, spatial-temporal bifurcation for KP equation is considered, several types of exact solutions to KP equation were constructed by bilinear form and homoclinic test approach. It is explicitly analyzed that the feature of the solitary wave is different on the

for KPI and deflexion point of soliton for KPII. As for KPI, when the equilibrium *u*<sup>0</sup> varies

Whereas, the *y*-periodic solitary wave changes into *x* − *t*-periodic solitary wave for KPII, the propagation direction of periodic solitary wave occurs outstanding deflexion. In addition,

some new type of multi-wave solutions are obtained using three-wave method.

Kadomtsev-Petviashvili (KP) equation in normalized variable *u*(*x*, *y*, *t*) reads

*uxt* <sup>−</sup> *uxxxx* <sup>−</sup> <sup>3</sup>(*u*2)*xx* <sup>−</sup> *<sup>s</sup>*

Let *u* = *u*<sup>0</sup> − 2(ln *F*)*ξξ*, Eq.(46) can be transformed into the following bilinear form

*<sup>y</sup>* <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0)*D*<sup>2</sup>

*F* = 1 + *b*1(*eip<sup>ξ</sup>* + *e*

2*p*2[4*b*<sup>2</sup>

Using "homoclinic test technique", we are going to seek the solution of the form

where *u* : *Rx* × *Ry* × *Rt* → *R*. It is easily to note that *u* = *u*<sup>0</sup> is an equilibrium solution of KP

*<sup>ξ</sup>* <sup>+</sup> *<sup>D</sup>*<sup>4</sup>

Substituting Eq.(49) into Eq.(48) with Eq.(47) yields the exact solution of Eq.(47) of the form

<sup>−</sup>*ip<sup>ξ</sup>* )*e*Ω*y*+*<sup>γ</sup>* + *b*2*e*

<sup>1</sup> <sup>+</sup> *<sup>b</sup>*1(*eip<sup>ξ</sup>* <sup>+</sup> *<sup>e</sup>*−*ip<sup>ξ</sup>* )(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]

**4.1. Spatiotemporal bifurcation and deflexion of the soliton**

<sup>6</sup> to another side, two-solitary wave changes into doubly periodic wave.

*uxt* <sup>−</sup> *uxxxx* <sup>−</sup> <sup>3</sup>(*u*2)*xx* <sup>−</sup> *uyy* <sup>=</sup> <sup>0</sup> (46)

*uyy* <sup>−</sup> *<sup>u</sup>ξξ* <sup>−</sup> <sup>3</sup>(*u*2)*ξξ* <sup>+</sup> *<sup>u</sup>ξξξξ* <sup>=</sup> <sup>0</sup> (47)

[*b*1(*eip<sup>ξ</sup>* <sup>+</sup> *<sup>e</sup>*−*ip<sup>ξ</sup>* )+(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]<sup>2</sup> (50)

<sup>6</sup> , which is a unique periodic bifurcation point

<sup>2</sup>*uyy* = 0 (45)

*<sup>ξ</sup>* − *A*]*F* · *F* = 0 (48)

<sup>2</sup>Ω*y*+2*<sup>γ</sup>* (49)

27].

from one side of <sup>−</sup><sup>1</sup>

both sides of equilibrium solution *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

equation, where *u*<sup>0</sup> is an arbitrary constant.

[*D*<sup>2</sup>

Now, we consider KPI equation

where *A*, *p*, Ω, *γ*, *b*<sup>1</sup> and *b*<sup>2</sup> are all real.

*u* = *u*<sup>0</sup> +

Setting *ξ* = *i*(*x* − *t*) gives

$$\begin{cases} u\_0 < -\frac{1}{6} \\ p^2 < -\frac{1 + 6u\_0}{4} \\ \Omega^2 = -p^4 - (1 + 6u\_0)p^2 \\ b\_1^2 = \frac{\Omega^2 b\_2}{\Omega^2 - 3p^4} \end{cases} \tag{53}$$

**Figure 4.** The two-soliton solution for KPI equation as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> 4

KPII equation is given by

$$
\hbar \mu\_{\rm xt} - \mu\_{\rm xxxx} - \Im (\mu^2)\_{\rm xx} + \mu\_{\rm yy} = 0 \tag{54}
$$

Making a variable transformation *ξ* = *x* − *t* in Eq.(54), it can be transformed into the following form

$$
u\_{yy} - 
u\_{\tilde{\xi}\tilde{\xi}} - \Im(\mu^2)\_{\tilde{\xi}\tilde{\xi}} - 
u\_{\tilde{\xi}\tilde{\xi}\tilde{\xi}\tilde{\xi}} = 0\tag{55}$$

Letting *u* = *u*<sup>0</sup> + 2(ln *F*)*ξξ* and using a similar way dealing with KPI, we take

$$F = 1 + b\_1(e^{ip\xi} + e^{-ip\xi})e^{\Omega y + \gamma} + b\_2e^{2\Omega y + 2\gamma}$$

By computing, the exact solution of Eq.(55) is given by

$$u = u\_0 - \frac{2p^2[4b\_1^2 + b\_1(e^{ip\_\sharp^x} + e^{-ip\_\sharp^x})(b\_2e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]}{[b\_1(e^{ip\_\sharp^x} + e^{-ip\_\sharp^x}) + (b\_2e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]^2} \tag{56}$$

#### 12 Will-be-set-by-IN-TECH 116 Wave Processes in Classical and New Solids Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations <sup>13</sup>

where parameters satisfy

$$
\Omega^2 = p^4 - (1 + 6u\_0)p^2 \qquad \qquad b\_1^2 = \frac{\Omega^2 b\_2}{\Omega^2 + 3p^4} \tag{57}
$$

It is easily to see that *<sup>u</sup>*<sup>0</sup> ≥ −<sup>1</sup> <sup>6</sup> is available as long as *<sup>p</sup>*<sup>2</sup> <sup>≥</sup> <sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0. Taking *ξ* = *x* − *t* into Eq.(56), the exact solution to KPII equation is expressed as

$$u\_2(\mathbf{x}, y, t) = u\_0 - \frac{2p^2[4b\_1^2 + b\_1 \cos(p(\mathbf{x} - t))(b\_2 e^{\Omega y + \gamma} + e^{-\Omega y - \gamma})]}{[b\_1 \cos(p(\mathbf{x} - t) + (b\_2 e^{\Omega y + \gamma} + e^{-\Omega y - \gamma}))]^2} \tag{58}$$

where

$$\begin{cases} u\_0 \ge -\frac{1}{6} \\ p^2 \ge 1 + 6u\_0 \\ \Omega^2 = p^4 - (1 + 6u\_0)p^2 \\ b\_1^2 = \frac{\Omega^2 b\_2}{\Omega^2 + 3p^4} \end{cases} \tag{59}$$


u

where

of <sup>−</sup><sup>1</sup>

Let

0

x

5

(a)

10


(b) The doubly periodic solution for KPI equation with *<sup>y</sup>* direction as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩


**Figure 7.** The *<sup>y</sup>*-periodic soliton solution for KPII equation as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

According to above discussion, we get that *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

KPII are interchanged around *u*0.

**4.2. Exact multi-wave solution**

20

u


*b*2

y

**Figure 6.** (a) The doubly periodic solution for KPI equation with *<sup>x</sup>* <sup>−</sup> *<sup>t</sup>* direction as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

*<sup>u</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>−</sup><sup>1</sup> 6 *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>1+6*u*<sup>0</sup> 4 <sup>Ω</sup><sup>2</sup> <sup>=</sup> <sup>−</sup>*p*<sup>4</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0)*p*<sup>2</sup>

<sup>1</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup> *<sup>b</sup>*<sup>2</sup> Ω<sup>2</sup>−3*p*<sup>4</sup>

The solution given by Eq.(60) represents periodic soliton with *y*-direction (see Fig.7).

0

x

10

KPI and deflexion of soliton for KPII. Around the both sides at *u*0, the property of solutions to KPI and KPII is all changed. As for KPI, when the equilibrium *u*<sup>0</sup> varies from one side

<sup>6</sup> to another side, two-soliton solution changes into doubly periodic solution. Whereas, the *y*-periodic soliton changes into *x*-periodic soliton for KPII. The double-soliton waves and doubly periodic soliton waves of KPI, periodic soliton waves on different spatial variable of

20


0

50 y

4

*u* = 2(ln *f*)*xx* (63)

<sup>6</sup> is a unique periodic bifurcation point for


u

0

0.5

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 117

(b)

x -10

8

1

0

8

(62)

10 20

y

Obviously, cos *p*(*x* − *t*) is periodic, so the solution given by Eq.(58) is a periodic soliton solution with *x* − *t*-direction (see Fig.5).


**Figure 5.** The *<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*-periodic soliton solution for KPII equation as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> 8


20


Comparing Eq.(47) with Eq.(55), it is easily to find that the Eq.(47) may be changed into Eq.(55) and vice versa by using the temporal and spatial transformation (*ξ*, *y*) → (*iξ*, *iy*). Because the solutions of KP equation are real functions, it is naturally to take specially *b*<sup>2</sup> = 1, *γ* = 0. Making variable transformation *ξ* → *iξ*, *y* → *iy*, *i* <sup>2</sup> <sup>=</sup> <sup>−</sup>1 in Eq.(50) and Eq.(56) yields

$$u\_3(\mathbf{x}, y, t) = u\_0 - \frac{2p^2[b\_1^2 + b\_1 \cos(p(\mathbf{x} - t)) \cos(\Omega y)]}{[b\_1 \cos(p(\mathbf{x} - t)) + \cos(\Omega y)]^2} \tag{60}$$

0 10 20

y

It is noted that the solution given by Eq.(59) is a singular periodic solution to KPI equation. In order to avoid the singularity, we set cos(*p*(*x* − *t*)) > 0 and cos(Ω*y*) > 0 (see Fig.6).

Besides, the *y*-periodic soliton solution to KPII is also given by

$$u\_4(x,y,t) = u\_0 + \frac{2p^2[4b\_1^2 + b\_1(e^{p(\mathbf{x}-t)} + e^{-p(\mathbf{x}-t)})(e^{i\Omega y} + e^{-i\Omega y})]}{[b\_1(e^{p(\mathbf{x}-t)} + e^{-p(\mathbf{x}-t)}) + (e^{i\Omega y} + e^{-i\Omega y})]^2} \tag{61}$$

**Figure 6.** (a) The doubly periodic solution for KPI equation with *<sup>x</sup>* <sup>−</sup> *<sup>t</sup>* direction as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> 8 (b) The doubly periodic solution for KPI equation with *<sup>y</sup>* direction as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> 8

where

12 Will-be-set-by-IN-TECH

<sup>6</sup> is available as long as *<sup>p</sup>*<sup>2</sup> <sup>≥</sup> <sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0.

<sup>Ω</sup><sup>2</sup> <sup>=</sup> *<sup>p</sup>*<sup>4</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0)*p*<sup>2</sup>

Obviously, cos *p*(*x* − *t*) is periodic, so the solution given by Eq.(58) is a periodic soliton

<sup>1</sup> <sup>=</sup> <sup>Ω</sup>2*b*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>b</sup>*1*cos*(*p*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*))(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]

[*b*1*cos*(*p*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)+(*b*2*e*Ω*y*+*<sup>γ</sup>* <sup>+</sup> *<sup>e</sup>*−Ω*y*−*γ*)]<sup>2</sup> (58)

<sup>Ω</sup><sup>2</sup> <sup>+</sup> <sup>3</sup>*p*<sup>4</sup> (57)

(59)

<sup>Ω</sup><sup>2</sup> <sup>=</sup> *<sup>p</sup>*<sup>4</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> <sup>6</sup>*u*0)*p*<sup>2</sup> *<sup>b</sup>*<sup>2</sup>

Taking *ξ* = *x* − *t* into Eq.(56), the exact solution to KPII equation is expressed as

*<sup>u</sup>*<sup>0</sup> ≥ −<sup>1</sup> 6 *<sup>p</sup>*<sup>2</sup> <sup>≥</sup> <sup>1</sup> <sup>+</sup> <sup>6</sup>*u*<sup>0</sup>

<sup>1</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup> *<sup>b</sup>*<sup>2</sup> Ω<sup>2</sup>+3*p*<sup>4</sup>

⎧ ⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩


20

**Figure 5.** The *<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*-periodic soliton solution for KPII equation as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

*<sup>u</sup>*3(*x*, *<sup>y</sup>*, *<sup>t</sup>*) = *<sup>u</sup>*<sup>0</sup> <sup>−</sup> <sup>2</sup>*p*2[*b*<sup>2</sup>

Besides, the *y*-periodic soliton solution to KPII is also given by

2*p*2[4*b*<sup>2</sup>


Making variable transformation *ξ* → *iξ*, *y* → *iy*, *i*

*u*4(*x*, *y*, *t*) = *u*<sup>0</sup> +


0

x

10

Comparing Eq.(47) with Eq.(55), it is easily to find that the Eq.(47) may be changed into Eq.(55) and vice versa by using the temporal and spatial transformation (*ξ*, *y*) → (*iξ*, *iy*). Because the solutions of KP equation are real functions, it is naturally to take specially *b*<sup>2</sup> = 1, *γ* = 0.

It is noted that the solution given by Eq.(59) is a singular periodic solution to KPI equation. In

<sup>1</sup> <sup>+</sup> *<sup>b</sup>*1(*ep*(*x*−*t*) <sup>+</sup> *<sup>e</sup>*−*p*(*x*−*t*)

order to avoid the singularity, we set cos(*p*(*x* − *t*)) > 0 and cos(Ω*y*) > 0 (see Fig.6).

20


<sup>1</sup> + *b*<sup>1</sup> cos(*p*(*x* − *t*)) cos(Ω*y*)]

y

8

<sup>2</sup> <sup>=</sup> <sup>−</sup>1 in Eq.(50) and Eq.(56) yields

[*b*<sup>1</sup> cos(*p*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)) + cos(Ω*y*)]<sup>2</sup> (60)

[*b*1(*ep*(*x*−*t*) <sup>+</sup> *<sup>e</sup>*−*p*(*x*−*<sup>t</sup>*))+(*ei*Ω*<sup>y</sup>* <sup>+</sup> *<sup>e</sup>*−*i*Ω*y*)]<sup>2</sup> (61)

)(*ei*Ω*<sup>y</sup>* + *e*−*i*Ω*y*)]

*b*2

where parameters satisfy

It is easily to see that *<sup>u</sup>*<sup>0</sup> ≥ −<sup>1</sup>

solution with *x* − *t*-direction (see Fig.5).

where

*<sup>u</sup>*2(*x*, *<sup>y</sup>*, *<sup>t</sup>*) = *<sup>u</sup>*<sup>0</sup> <sup>−</sup> <sup>2</sup>*p*2[4*b*<sup>2</sup>

$$\begin{cases} u\_0 < -\frac{1}{6} \\ p^2 < -\frac{1 + 6u\_0}{4} \\ \Omega^2 = -p^4 - (1 + 6u\_0)p^2 \\ b\_1^2 = \frac{\Omega^2 b\_2}{\Omega^2 - 3p^4} \end{cases} \tag{62}$$

The solution given by Eq.(60) represents periodic soliton with *y*-direction (see Fig.7).

**Figure 7.** The *<sup>y</sup>*-periodic soliton solution for KPII equation as *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> 4

According to above discussion, we get that *<sup>u</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>6</sup> is a unique periodic bifurcation point for KPI and deflexion of soliton for KPII. Around the both sides at *u*0, the property of solutions to KPI and KPII is all changed. As for KPI, when the equilibrium *u*<sup>0</sup> varies from one side of <sup>−</sup><sup>1</sup> <sup>6</sup> to another side, two-soliton solution changes into doubly periodic solution. Whereas, the *y*-periodic soliton changes into *x*-periodic soliton for KPII. The double-soliton waves and doubly periodic soliton waves of KPI, periodic soliton waves on different spatial variable of KPII are interchanged around *u*0.

#### **4.2. Exact multi-wave solution**

Let

$$
\mu = \Im(\ln f)\_{\text{xx}} \tag{63}
$$

#### 14 Will-be-set-by-IN-TECH 118 Wave Processes in Classical and New Solids Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations <sup>15</sup>

in Eq.(45), where *f* = *f*(*x*, *y*, *t*) is an unknown real function. Substituting Eq.(63) into Eq.(45), we can reduce Eq.(45) into the bilinear form

$$(D\_x D\_t - D\_x^4 - p^2 D\_y^2) f \cdot f = 0 \tag{64}$$

Similarity, the breather two-solitary wave solutions of KPII equation also obtained as follows

<sup>√</sup>*a*1*a*−1(*A*<sup>2</sup>

where *<sup>γ</sup>*<sup>3</sup> = *<sup>p</sup>*1(*<sup>x</sup>* − *<sup>C</sup>*2*t*), *<sup>δ</sup>*<sup>3</sup> = *<sup>A</sup>*2(*<sup>x</sup>* + *<sup>D</sup>*2*<sup>y</sup>* + *<sup>B</sup>*2*t*) + *<sup>θ</sup>*<sup>3</sup> and *<sup>ξ</sup>*<sup>3</sup> = *<sup>p</sup>*1(*<sup>x</sup>* + *<sup>β</sup>*2*<sup>y</sup>* + *<sup>C</sup>*2*t*) as *<sup>a</sup>*1*a*−<sup>1</sup> >

<sup>2</sup> <sup>+</sup> <sup>96</sup>*p*<sup>4</sup> 1

(cos(*γ*3) + 2

<sup>2</sup> <sup>−</sup> <sup>4</sup>*p*<sup>2</sup> 1)*β*<sup>2</sup>

(cos(*γ*4) + 2

2*β*<sup>2</sup> 2

<sup>√</sup>*a*1*a*−1(*p*<sup>2</sup>

where *γ*<sup>4</sup> = *p*1(*x* − *C*3*t*), *δ*<sup>4</sup> = *p*3(*x* + *D*3*y* + *B*3*t*) + *θ*3, *ξ*<sup>4</sup> = *p*1(*x* + *A*3*y* + *C*3*t*), with

<sup>3</sup>), *<sup>C</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>2(3*p*<sup>2</sup>

**5. Kadomtsev-Petviashvili equation with positive dispersion**

*<sup>F</sup>*(*Dx*, *Dy*, *Dt*)*<sup>f</sup>* · *<sup>f</sup>* = (*DxDt* <sup>+</sup> *<sup>D</sup>*<sup>4</sup>

<sup>√</sup>*a*1*a*−1((*A*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>3</sup> <sup>+</sup> *<sup>ξ</sup>*3) <sup>−</sup> (*A*<sup>2</sup> <sup>+</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>3</sup> <sup>−</sup> *<sup>ξ</sup>*3))

<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*3) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*3))<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*3) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*3))<sup>2</sup>

<sup>√</sup>*a*1*a*−1*A*<sup>2</sup> sinh(*δ*3)) sin(*γ*3) <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

, *<sup>C</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>4*p*<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*4) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*4))<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> *<sup>p</sup>*<sup>3</sup> sinh(*δ*4)) sin(*γ*4) <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*4) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*4))<sup>2</sup> )

<sup>1</sup>), *D*<sup>3</sup> =

<sup>1</sup>) cosh(*δ*4) cos(*γ*4)

<sup>√</sup>3(*p*<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*3) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*3))<sup>2</sup> )

<sup>√</sup>*a*1*a*−1((*p*<sup>3</sup> <sup>−</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>4</sup> <sup>+</sup> *<sup>ξ</sup>*4) <sup>−</sup> (*p*<sup>3</sup> <sup>+</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>4</sup> <sup>−</sup> *<sup>ξ</sup>*4))

<sup>3</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*4) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*4))<sup>2</sup>

<sup>3</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

The purpose of this section is to investigate the fission and fusion interactions of the breather-type multi-solitary waves solutions to the KP equation with positive dispersion. By transformation of independent variable *<sup>t</sup>* → −*t*, *<sup>y</sup>* <sup>→</sup> <sup>√</sup>3*<sup>y</sup>* in (\*), KP equation with positive

*ut* <sup>+</sup> <sup>6</sup>*uux* <sup>+</sup> *uxxx* <sup>−</sup> <sup>3</sup>*∂*−<sup>1</sup>

<sup>1</sup>) cosh(*δ*3) cos(*γ*3)

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 119

1

1

, *θ*<sup>3</sup> = ln(

 *a*<sup>1</sup> *a*−<sup>1</sup> )

<sup>3</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup> 1)

*<sup>x</sup> uyy* = 0 (66)

*<sup>y</sup>*)*f* · *f* = 0 (68)

*u* = 2(ln *f*)*xx* (67)

*<sup>x</sup>* <sup>−</sup> <sup>3</sup>*D*<sup>2</sup>

*p*3

<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> 2

2*β*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> 2 <sup>2</sup> , *<sup>D</sup>*<sup>2</sup> <sup>=</sup> <sup>12</sup>*p*<sup>4</sup>

*u*3(*x*, *y*, *t*) = 2(

0. Here

*A*<sup>2</sup> =

*A*<sup>3</sup> = 2 √

 2*p*<sup>2</sup> 1*β*2 *a*2

*<sup>a</sup>*1*a*−1*A*<sup>2</sup>

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 2 *p*2 <sup>1</sup> + 2

(cos(*γ*3) + 2

2*p*1(*a*<sup>2</sup> *p*<sup>1</sup> cosh(*ξ*3) + 2

(cos(*γ*3) + 2

, *<sup>B</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> (*β*<sup>2</sup>

<sup>3</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 2 *p*2 <sup>1</sup> + 2

(cos(*γ*4) + 2

2*p*1(*a*<sup>2</sup> *p*<sup>1</sup> cosh(*ξ*4) + 2

(cos(*γ*4) + 2

<sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

+

+

<sup>2</sup> <sup>−</sup> <sup>12</sup>*p*<sup>4</sup> 1

*a*−1, *a*2, *p*<sup>1</sup> and *β*<sup>2</sup> are free parameters. And

+

+

<sup>3</sup>*p*3, *<sup>B</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>2(3*p*<sup>2</sup>

*a*−1, *a*2, *p*<sup>1</sup> and *p*<sup>3</sup> are free parameters.

By the transformation of a dependent variable *u*

Eq.(66) can be transformed into the bilinear form

dispersion can be written as

where *f*(*x*, *y*, *t*) is a real function.

*a*2

<sup>4</sup>*a*1*a*−<sup>1</sup> *<sup>p</sup>*<sup>2</sup>

*β*2

*u*4(*x*, *y*, *t*) = 2(

In order to obtain three wave solution of KP equation, we provide that

$$f = \cos(\gamma\_1) + a\_{-1} \exp(-\delta\_1) + a\_1 \exp(\delta\_1) + a\_2 \sinh(\tilde{\xi}\_1) \tag{65}$$

where *γ*<sup>1</sup> = *p*1(*x* − *α*1*t*), *δ*<sup>1</sup> = *p*3(*x* + *β*3*y* + *α*3*t*) and *ξ*<sup>1</sup> = *p*2(*x* + *β*2*y* + *α*2*t*). Substituting Eq.(65) into Eq.(64)and equating the coefficients of all powers of sinh(*ξ*1), cos(*γ*1), sinh(*ξ*1), exp(*δ*1), sinh(*ξ*1) exp(−*δ*1), cosh(*ξ*1), sin(*γ*1), cosh(*ξ*1) exp(*δ*1), cosh(*ξ*1) exp(−*δ*1), sin(*γ*1) exp(*δ*1), sin(*γ*1) exp(−*δ*1), cos(*γ*1) exp(*δ*1), cos(*γ*1) exp(−*δ*1) to zero, we can obtain a set of algebraic equations for *a*−1, *a*2, *a*1, *α*1, *α*2, *α*3, *β*2, *β*3, *p*1, *p*<sup>2</sup> and *p*3, then by solving these set of algebraic equations and let *p*<sup>1</sup> = *p*2, we obtain the breather two-solitary wave (three wave) solution of KPI as follows:

$$\begin{split} u\_{1}(x,y,t) &= 2(\frac{a\_{2}\sqrt{a\_{1}a\_{-1}}((A\_{1}-p\_{1})^{2}\sinh(\delta\_{2}+\xi\_{2})-(A\_{1}+p\_{1})^{2}\sinh(\delta\_{2}-\xi\_{2}))}{(\cos(\gamma\_{2})+2\sqrt{a\_{1}a\_{-1}}\cosh(\delta\_{2})+a\_{2}\sinh(\xi\_{2}))^{2}} \\ &+ \frac{4a\_{1}a\_{-1}A\_{1}^{2}-a\_{2}^{2}p\_{1}^{2}+2\sqrt{a\_{1}a\_{-1}}(A\_{1}^{2}-p\_{1}^{2})\cosh(\delta\_{2})\cos(\gamma\_{2})}{(\cos(\gamma\_{2})+2\sqrt{a\_{1}a\_{-1}}\cosh(\delta\_{2})+a\_{2}\sinh(\xi\_{2}))^{2}} \\ &+ \frac{2p\_{1}(a\_{2}p\_{1}\cosh(\tilde{\xi}\_{2})+2\sqrt{a\_{1}a\_{-1}}A\_{1}\sinh(\delta\_{2}))\sin(\gamma\_{2})-p\_{1}^{2}}{(\cos(\gamma\_{2})+2\sqrt{a\_{1}a\_{-1}}\cosh(\delta\_{2})+a\_{2}\sinh(\tilde{\xi}\_{2}))^{2}} \end{split}$$

where

$$\begin{aligned} \alpha\_1 a\_{-1} &> 0, & \theta\_1 &= \ln(\sqrt{\frac{a\_1}{a\_{-1}}}), & \gamma\_2 &= p\_1(\mathbf{x} - \mathbf{C}\_1 t) \\ \delta\_2 &= A\_1(\mathbf{x} + D\_1 y + B\_1 t) + \theta\_1, & \tilde{\xi}\_2 &= p\_1(\mathbf{x} + \beta\_2 y + \mathbf{C}\_1 t) \\ A\_1 &= \frac{\sqrt{2p\_1^2 \beta\_2^2 + 12p\_1^4}}{\beta\_2}, & B\_1 &= -\frac{(4p\_1^2 + \beta\_2^2)\beta\_2^2 + 96p\_1^4}{2\beta\_2^2} \\ \mathcal{C}\_1 &= -\frac{4p\_1^2 - \beta\_2^2}{2}, & D\_1 &= -\frac{12p\_1^4 - \beta\_2^2}{2\beta\_2} \end{aligned}$$

Here, *a*−1, *a*2, *p*<sup>1</sup> and *β*<sup>2</sup> are free parameters.

In the case of *<sup>a</sup>*1*a*−<sup>1</sup> < 0, the breather two-solitary wave solution of KPI can be expressed as

$$\begin{split} u\_{2}(\mathbf{x},y,t) &= 2(\frac{a\_{2}\sqrt{\gamma}((A\_{1}-p\_{1})^{2}\cosh(\delta\_{2}+\tilde{\xi}\_{2})-(A\_{1}+p\_{1})^{2}\cosh(\delta\_{2}-\tilde{\xi}\_{2}))+4\gamma A\_{1}^{2}-a\_{2}^{2}p\_{1}^{2}) \\ &+ \frac{2\sqrt{\gamma}(A\_{1}^{2}-p\_{1}^{2})\sinh(\delta\_{2})\cos(\gamma\_{2})+2p\_{1}(a\_{2}p\_{1}\cosh(\tilde{\xi}\_{2})+2\sqrt{\gamma}A\_{1}\cosh(\delta\_{2}))\sin(\gamma\_{2})-p\_{1}^{2})}{(\cos(\gamma\_{2})+\sqrt{\gamma}\sinh(\delta\_{2})+a\_{2}\sinh(\tilde{\xi}\_{2}))^{2}} \\ \text{where } a=-a\_{2}, a\_{1} \end{split}$$

where *<sup>γ</sup>* = −*a*1*a*−1.

Similarity, the breather two-solitary wave solutions of KPII equation also obtained as follows

$$\begin{split} u\_{3}(x,y,t) &= 2(\frac{a\_{2}\sqrt{a\_{1}a\_{-}}((A\_{2}-p\_{1})^{2}\sinh(\delta\_{3}+\tilde{\xi}\_{3})-(A\_{2}+p\_{1})^{2}\sinh(\delta\_{3}-\tilde{\xi}\_{3}))}{(\cos(\gamma\_{3})+2\sqrt{a\_{1}a\_{-}}\cosh(\delta\_{3})+a\_{2}\sinh(\tilde{\xi}\_{3}))^{2}} \\ &+\frac{a\_{1}a\_{-1}A\_{2}^{2}-a\_{2}^{2}p\_{1}^{2}+2\sqrt{a\_{1}a\_{-1}}(A\_{2}^{2}-p\_{1}^{2})\cosh(\delta\_{3})\cos(\gamma\_{3})}{(\cos(\gamma\_{3})+2\sqrt{a\_{1}a\_{-1}}\cosh(\delta\_{3})+a\_{2}\sinh(\tilde{\xi}\_{3}))^{2}} \\ &+\frac{2p\_{1}(a\_{2}p\_{1}\cosh(\tilde{\xi}\_{3})+2\sqrt{a\_{1}a\_{-1}}A\_{2}\sinh(\delta\_{3}))\sin(\gamma\_{3})-p\_{1}^{2}}{(\cos(\gamma\_{3})+2\sqrt{a\_{1}a\_{-1}}\cosh(\delta\_{3})+a\_{2}\sinh(\tilde{\xi}\_{3}))^{2}} \end{split}$$

where *<sup>γ</sup>*<sup>3</sup> = *<sup>p</sup>*1(*<sup>x</sup>* − *<sup>C</sup>*2*t*), *<sup>δ</sup>*<sup>3</sup> = *<sup>A</sup>*2(*<sup>x</sup>* + *<sup>D</sup>*2*<sup>y</sup>* + *<sup>B</sup>*2*t*) + *<sup>θ</sup>*<sup>3</sup> and *<sup>ξ</sup>*<sup>3</sup> = *<sup>p</sup>*1(*<sup>x</sup>* + *<sup>β</sup>*2*<sup>y</sup>* + *<sup>C</sup>*2*t*) as *<sup>a</sup>*1*a*−<sup>1</sup> > 0. Here

$$A\_2 = \frac{\sqrt{2p\_1^2 \beta\_2^2 - 12p\_1^4}}{\beta\_2}, \; B\_2 = -\frac{(\beta\_2^2 - 4p\_1^2)\beta\_2^2 + 96p\_1^4}{2\beta\_2^2}, \; C\_2 = -\frac{4p\_1^2 + \beta\_2^2}{2}, \; D\_2 = \frac{12p\_1^4 + \beta\_2^2}{2\beta\_2}$$

*a*−1, *a*2, *p*<sup>1</sup> and *β*<sup>2</sup> are free parameters. And

14 Will-be-set-by-IN-TECH

in Eq.(45), where *f* = *f*(*x*, *y*, *t*) is an unknown real function. Substituting Eq.(63) into Eq.(45),

*<sup>x</sup>* <sup>−</sup> *<sup>p</sup>*2*D*<sup>2</sup>

where *γ*<sup>1</sup> = *p*1(*x* − *α*1*t*), *δ*<sup>1</sup> = *p*3(*x* + *β*3*y* + *α*3*t*) and *ξ*<sup>1</sup> = *p*2(*x* + *β*2*y* + *α*2*t*). Substituting Eq.(65) into Eq.(64)and equating the coefficients of all powers of sinh(*ξ*1), cos(*γ*1), sinh(*ξ*1), exp(*δ*1), sinh(*ξ*1) exp(−*δ*1), cosh(*ξ*1), sin(*γ*1), cosh(*ξ*1) exp(*δ*1), cosh(*ξ*1) exp(−*δ*1), sin(*γ*1) exp(*δ*1), sin(*γ*1) exp(−*δ*1), cos(*γ*1) exp(*δ*1), cos(*γ*1) exp(−*δ*1) to zero, we can obtain a set of algebraic equations for *a*−1, *a*2, *a*1, *α*1, *α*2, *α*3, *β*2, *β*3, *p*1, *p*<sup>2</sup> and *p*3, then by solving these set of algebraic equations and let *p*<sup>1</sup> = *p*2, we obtain the breather two-solitary wave (three wave)

<sup>√</sup>*a*1*a*−1(*A*<sup>2</sup>

 *a*<sup>1</sup> *a*−<sup>1</sup>

<sup>2</sup> , *<sup>D</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>12*p*<sup>4</sup>

<sup>1</sup>) sinh(*δ*2) cos(*γ*2) + 2*p*1(*a*<sup>2</sup> *p*<sup>1</sup> cosh(*ξ*2) + 2

In the case of *<sup>a</sup>*1*a*−<sup>1</sup> < 0, the breather two-solitary wave solution of KPI can be expressed as

*δ*<sup>2</sup> = *A*1(*x* + *D*1*y* + *B*1*t*) + *θ*1, *ξ*<sup>2</sup> = *p*1(*x* + *β*2*y* + *C*1*t*)

, *<sup>B</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> (4*p*<sup>2</sup>

*<sup>f</sup>* = cos(*γ*1) + *<sup>a</sup>*−<sup>1</sup> exp(−*δ*1) + *<sup>a</sup>*<sup>1</sup> exp(*δ*1) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*1) (65)

<sup>√</sup>*a*1*a*−1((*A*<sup>1</sup> <sup>−</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>2</sup> <sup>+</sup> *<sup>ξ</sup>*2) <sup>−</sup> (*A*<sup>1</sup> <sup>+</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2))

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*2) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*2))<sup>2</sup>

<sup>√</sup>*a*1*a*−1*A*<sup>1</sup> sinh(*δ*2)) sin(*γ*2) <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

), *γ*<sup>2</sup> = *p*1(*x* − *C*1*t*)

<sup>1</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> 2)*β*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> 2

2*β*<sup>2</sup>

<sup>√</sup>*γ*((*A*<sup>1</sup> <sup>−</sup> *<sup>p</sup>*1)<sup>2</sup> cosh(*δ*<sup>2</sup> <sup>+</sup> *<sup>ξ</sup>*2) <sup>−</sup> (*A*<sup>1</sup> <sup>+</sup> *<sup>p</sup>*1)<sup>2</sup> cosh(*δ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2)) + <sup>4</sup>*γA*<sup>2</sup>

(cos(*γ*2) + <sup>√</sup>*<sup>γ</sup>* sinh(*δ*2) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*2))<sup>2</sup>

(cos(*γ*2) + <sup>√</sup>*<sup>γ</sup>* sinh(*δ*2) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*2))<sup>2</sup> )

2*β*<sup>2</sup> 2

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*2) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*2))<sup>2</sup> )

<sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*2) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*2))<sup>2</sup>

<sup>1</sup>) cosh(*δ*2) cos(*γ*2)

1

<sup>2</sup> <sup>+</sup> <sup>96</sup>*p*<sup>4</sup> 1

<sup>√</sup>*γA*<sup>1</sup> cosh(*δ*2)) sin(*γ*2) <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 2 *p*2 1

1

*<sup>y</sup>*)*f* · *f* = 0 (64)

(*DxDt* <sup>−</sup> *<sup>D</sup>*<sup>4</sup>

(cos(*γ*2) + 2

In order to obtain three wave solution of KP equation, we provide that

we can reduce Eq.(45) into the bilinear form

solution of KPI as follows:

where

*u*2(*x*, *y*, *t*) = 2(

where *<sup>γ</sup>* = −*a*1*a*−1.

+ 2 √*γ*(*A*<sup>2</sup>

*u*1(*x*, *y*, *t*) = 2(

+

+

*A*<sup>1</sup> =

Here, *a*−1, *a*2, *p*<sup>1</sup> and *β*<sup>2</sup> are free parameters.

*a*2

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

 2*p*<sup>2</sup> 1*β*2

*<sup>C</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>4*p*<sup>2</sup>

*a*2

<sup>4</sup>*a*1*a*−1*A*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 2 *p*2 <sup>1</sup> + 2

(cos(*γ*2) + 2

2*p*1(*a*<sup>2</sup> *p*<sup>1</sup> cosh(*ξ*2) + 2

(cos(*γ*2) + 2

*<sup>a</sup>*1*a*−<sup>1</sup> > 0, *<sup>θ</sup>*<sup>1</sup> = ln(

<sup>2</sup> <sup>+</sup> <sup>12</sup>*p*<sup>4</sup> 1

*β*2

<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> 2

*u*4(*x*, *y*, *t*) = 2( *a*2 <sup>√</sup>*a*1*a*−1((*p*<sup>3</sup> <sup>−</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>4</sup> <sup>+</sup> *<sup>ξ</sup>*4) <sup>−</sup> (*p*<sup>3</sup> <sup>+</sup> *<sup>p</sup>*1)<sup>2</sup> sinh(*δ*<sup>4</sup> <sup>−</sup> *<sup>ξ</sup>*4)) (cos(*γ*4) + 2 <sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*4) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*4))<sup>2</sup> + <sup>4</sup>*a*1*a*−<sup>1</sup> *<sup>p</sup>*<sup>2</sup> <sup>3</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 2 *p*2 <sup>1</sup> + 2 <sup>√</sup>*a*1*a*−1(*p*<sup>2</sup> <sup>3</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> <sup>1</sup>) cosh(*δ*4) cos(*γ*4) (cos(*γ*4) + 2 <sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*4) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*4))<sup>2</sup> + 2*p*1(*a*<sup>2</sup> *p*<sup>1</sup> cosh(*ξ*4) + 2 <sup>√</sup>*a*1*a*−<sup>1</sup> *<sup>p</sup>*<sup>3</sup> sinh(*δ*4)) sin(*γ*4) <sup>−</sup> *<sup>p</sup>*<sup>2</sup> 1 (cos(*γ*4) + 2 <sup>√</sup>*a*1*a*−<sup>1</sup> cosh(*δ*4) + *<sup>a</sup>*<sup>2</sup> sinh(*ξ*4))<sup>2</sup> ) where *γ*<sup>4</sup> = *p*1(*x* − *C*3*t*), *δ*<sup>4</sup> = *p*3(*x* + *D*3*y* + *B*3*t*) + *θ*3, *ξ*<sup>4</sup> = *p*1(*x* + *A*3*y* + *C*3*t*), with *A*<sup>3</sup> = 2 √ <sup>3</sup>*p*3, *<sup>B</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>2(3*p*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup> <sup>3</sup>), *<sup>C</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>2(3*p*<sup>2</sup> <sup>3</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup> <sup>1</sup>), *D*<sup>3</sup> = <sup>√</sup>3(*p*<sup>2</sup> <sup>3</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup> 1) , *θ*<sup>3</sup> = ln( *a*<sup>1</sup> )

*a*−1, *a*2, *p*<sup>1</sup> and *p*<sup>3</sup> are free parameters.

## **5. Kadomtsev-Petviashvili equation with positive dispersion**

The purpose of this section is to investigate the fission and fusion interactions of the breather-type multi-solitary waves solutions to the KP equation with positive dispersion.

By transformation of independent variable *<sup>t</sup>* → −*t*, *<sup>y</sup>* <sup>→</sup> <sup>√</sup>3*<sup>y</sup>* in (\*), KP equation with positive dispersion can be written as

$$
\mu\_t + 6\mu u\_x + u\_{xxx} - 3\partial\_x^{-1} u\_{yy} = 0 \tag{66}
$$

By the transformation of a dependent variable *u*

$$
\mu = \Im(\ln f)\_{\text{xx}} \tag{67}
$$

*p*3

*a*−<sup>1</sup>

Eq.(66) can be transformed into the bilinear form

$$F(D\_\mathbf{x}, D\_\mathbf{y}, D\_t)f \cdot f = (D\_\mathbf{x}D\_l + D\_\mathbf{x}^4 - 3D\_\mathbf{y}^2)f \cdot f = 0\tag{68}$$

where *f*(*x*, *y*, *t*) is a real function.

#### **5.1. Fission and fusion of multi-wave**

In the following, by using generalized three-wave type of Ans*atz* ¨ approach, we study the interaction and spatiotemporal feature of three-wave of KP equation with positive dispersion. Now we suppose the solution of Eq.(68) as

$$f\left(\mathbf{x}, y, t\right) = e^{\tilde{\xi}\_1} + \delta\_1 \cos \tilde{\xi}\_2 + \delta\_2 \cosh \tilde{\xi}\_3 + \delta\_3 e^{-\tilde{\xi}\_1} \tag{69}$$

Fig.8 is the process of interaction for two solitons solutions with the evolution of time, where *s*<sup>1</sup> and *s*<sup>2</sup> are the different two solitons, and *s*<sup>12</sup> represents the interaction between *s*1,*s*<sup>2</sup> and the periodic wave cos(*ξ*2). The phenomena of soliton interaction are clearly presented. It shows that the two soliton experience interaction, they will fusion with few oscillations and later travel ahead continuously. The value of *δ*<sup>2</sup> will determine the length of resonant soliton. Obviously, the solitons *s*<sup>1</sup> and *s*<sup>2</sup> can not approach each together closely from picture. They

Usually, the interactions between solitons for a lot of integrable or non-integrable system are considered to be completely elastic. That is to say, the amplitude, velocity and wave shape of a soliton do not change after the nonlinear collisions. However, for several nonlinear partial differential equations, completely nonelastic interactions will occur. On the other hand, two or more solitons will fusion to one soliton. These two types of phenomena was called soliton fission and soliton fusion, respectively. Now we demonstrate soliton fission and fusion of the

S\_1

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*2*b*1)*δ*<sup>2</sup>

, *a*<sup>1</sup> �= 0 and *a*<sup>2</sup> �= 0.

<sup>2</sup> <sup>+</sup> *<sup>b</sup>*2) <sup>−</sup> *<sup>a</sup>*2*b*1)*δ*<sup>2</sup>

<sup>2</sup> cos *ξ*<sup>2</sup> + *δ*2*a*<sup>3</sup>

*eξ*<sup>1</sup> + *δ*<sup>1</sup> cos *ξ*<sup>2</sup> + *δ*<sup>2</sup> cosh *ξ*<sup>3</sup>

*<sup>a</sup>*1*eξ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*1*a*<sup>2</sup> sin *<sup>ξ</sup>*<sup>2</sup> <sup>+</sup> *<sup>δ</sup>*2*a*<sup>3</sup> sinh *<sup>ξ</sup>*<sup>3</sup>

*eξ*<sup>1</sup> + *δ*<sup>1</sup> cos *ξ*<sup>2</sup> + *δ*<sup>2</sup> cosh *ξ*<sup>3</sup>

<sup>3</sup> + *a*2*b*<sup>1</sup> + *a*1*a*<sup>2</sup>

<sup>2</sup> can not be zero, then the condition must be satisfied:

<sup>2</sup> <sup>−</sup> (*a*<sup>2</sup>

<sup>2</sup> cosh *ξ*3)

2

2

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> 2) <sup>3</sup>*δ*<sup>2</sup> <sup>1</sup> = 0

–20

–10

y

**Figure 8.** The plot of the space structure of the breather-type multi-solitary waves solutions and contour

<sup>3</sup> + *a*1*b*<sup>2</sup> + *a*1*a*<sup>2</sup>

2

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

<sup>2</sup>*eξ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*1*a*<sup>2</sup>

<sup>2</sup> − *<sup>a</sup>*2*b*1)(*a*<sup>1</sup>

0

10

S\_2

S\_12

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 121

–30 –20 –10 0 10 20 30 x

<sup>2</sup> <sup>−</sup> (*a*<sup>1</sup>

<sup>2</sup> + *a*<sup>2</sup>

<sup>2</sup>)3*δ*<sup>1</sup> 2)

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*1*b*2) <sup>=</sup> <sup>0</sup>

(72)

S'\_2

S'\_1

interact produce the breather wave.

–30 –20 –10 0 10 20 30

Δ ((*a*<sup>1</sup>

where Δ = *a*<sup>2</sup>

For *a*<sup>2</sup> 2 *a*2 <sup>2</sup> + *a*<sup>1</sup>

x

<sup>3</sup> + *a*2*b*<sup>1</sup> + *a*1*a*<sup>2</sup>

<sup>2</sup> + *a*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

So, we obtain the following solution:

<sup>2</sup>)3(*a*<sup>1</sup>

<sup>2</sup><sup>2</sup> <sup>+</sup> (*a*2*b*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*2*a*1)

plot map of *u* in (*x*, *y*)-plane. In Eq.(70), let *δ*<sup>3</sup> = 0, then

4(*a*<sup>1</sup>

2 *a*2 <sup>2</sup> + *a*<sup>1</sup>

(*a*1(*a*<sup>2</sup>

Kadomtsev-Petviashvili equation with positive dispersion.

–30 –20 –10 0 10 20 30

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*1*b*2)(*a*<sup>1</sup>

<sup>3</sup> + *a*1*b*<sup>2</sup> + *a*1*a*<sup>2</sup>

<sup>2</sup><sup>2</sup> <sup>+</sup> (*a*2*b*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*2*a*1)

<sup>2</sup> <sup>−</sup> *<sup>b</sup>*2) + *<sup>a</sup>*2*b*1)(*a*1(*a*<sup>2</sup>

*<sup>u</sup>*(*x*, *<sup>y</sup>*, *<sup>t</sup>*) = <sup>2</sup>(*a*<sup>1</sup>

− 2 

y

where *ξ*<sup>1</sup> = *a*1*x* + *b*1*y* + *c*1*t* + *θ*1, *ξ*<sup>2</sup> = *a*2*x* + *b*2*y* + *c*2*t* + *θ*2, *ξ*<sup>3</sup> = *a*3*x* + *b*3*y* + *c*3*t* + *θ*<sup>3</sup> and *aj*, *bj*, *cj*, *δj*, *j* = 1, 2, 3 are some constants to be determined. Substituting the Ans*atz* ¨ Eq.(69) into Eq. (68) will produce the following relations:

*<sup>a</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup> *<sup>b</sup>*2*a*<sup>1</sup> <sup>−</sup> *<sup>a</sup>*2*b*<sup>1</sup> *a*2 <sup>2</sup> + *a*<sup>1</sup> <sup>2</sup> , *<sup>c</sup>*<sup>3</sup> <sup>=</sup> <sup>Δ</sup><sup>1</sup> (*a*<sup>2</sup> <sup>2</sup> + *a*<sup>1</sup> 2) 3 *<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>a</sup>*<sup>1</sup> <sup>5</sup> <sup>−</sup> <sup>2</sup> *<sup>a</sup>*<sup>1</sup> 3*a*2 <sup>2</sup> + 3 *b*<sup>2</sup> <sup>2</sup>*a*<sup>1</sup> <sup>−</sup> <sup>3</sup> *<sup>b</sup>*<sup>1</sup> <sup>2</sup>*a*<sup>1</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*<sup>2</sup> <sup>4</sup>*a*<sup>1</sup> <sup>−</sup> <sup>6</sup> *<sup>a</sup>*2*b*1*b*<sup>2</sup> *a*2 <sup>2</sup> + *a*<sup>1</sup> 2 *<sup>b</sup>*<sup>3</sup> <sup>=</sup> <sup>2</sup> *<sup>a</sup>*<sup>1</sup> 3*a*2 <sup>3</sup> + *a*<sup>2</sup> <sup>2</sup>*b*1*b*<sup>2</sup> + *a*<sup>1</sup> <sup>5</sup>*a*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>1</sup> <sup>2</sup>*b*1*b*<sup>2</sup> + *a*<sup>2</sup> <sup>5</sup>*a*<sup>1</sup> + *b*<sup>1</sup> <sup>2</sup>*a*1*a*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> <sup>2</sup>*a*1*a*<sup>2</sup> (*a*<sup>2</sup> <sup>2</sup> + *a*<sup>1</sup> 2) 2 *<sup>δ</sup>*<sup>3</sup> <sup>=</sup> <sup>Δ</sup><sup>2</sup> 4 (*a*<sup>2</sup> <sup>2</sup> + *a*<sup>1</sup> 2) <sup>3</sup> (*a*<sup>1</sup> <sup>3</sup> + *a*2*b*<sup>1</sup> + *a*1*a*<sup>2</sup> <sup>2</sup> − *<sup>b</sup>*2*a*1) (*a*<sup>1</sup> <sup>3</sup> + *b*2*a*<sup>1</sup> + *a*1*a*<sup>2</sup> <sup>2</sup> − *<sup>a</sup>*2*b*1) *<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>a</sup>*<sup>1</sup> <sup>4</sup>*a*<sup>2</sup> + 2 *a*<sup>1</sup> 2*a*2 <sup>3</sup> <sup>−</sup> <sup>3</sup> *<sup>b</sup>*<sup>2</sup> <sup>2</sup>*a*<sup>2</sup> + 3 *b*<sup>1</sup> <sup>2</sup>*a*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> <sup>5</sup> <sup>−</sup> <sup>6</sup> *<sup>b</sup>*1*b*2*a*<sup>1</sup> *a*2 <sup>2</sup> + *a*<sup>1</sup> 2 Δ<sup>1</sup> = 3 *b*2*a*<sup>1</sup> <sup>7</sup> + 3 *a*2*b*1*a*<sup>1</sup> <sup>6</sup> + 9 *a*<sup>2</sup> <sup>2</sup>*b*2*a*<sup>1</sup> <sup>5</sup> + 9 *a*<sup>2</sup> <sup>3</sup>*b*1*a*<sup>1</sup> <sup>4</sup> + 9 *a*<sup>1</sup> 3*a*2 <sup>4</sup>*b*<sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*<sup>1</sup> 3*b*1 2*b*2 +*a*<sup>1</sup> 3*b*2 <sup>3</sup> + 3 *a*<sup>1</sup> <sup>2</sup>*a*2*b*<sup>1</sup> <sup>3</sup> <sup>−</sup> <sup>9</sup> *<sup>a</sup>*<sup>1</sup> <sup>2</sup>*a*2*b*1*b*<sup>2</sup> <sup>2</sup> + 9 *a*<sup>1</sup> 2*a*2 <sup>5</sup>*b*<sup>1</sup> + 3 *a*1*a*<sup>2</sup> <sup>6</sup>*b*<sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*1*a*<sup>2</sup> 2*b*2 3 +9 *a*1*a*<sup>2</sup> 2*b*1 <sup>2</sup>*b*<sup>2</sup> + 3 *a*<sup>2</sup> <sup>3</sup>*b*1*b*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 3*b*1 <sup>3</sup> + 3 *a*<sup>2</sup> 7*b*1 Δ<sup>2</sup> = − *b*2 2*a*1 <sup>2</sup> <sup>−</sup> <sup>2</sup> *<sup>a</sup>*2*b*1*b*2*a*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> 2*b*1 <sup>2</sup> + *a*<sup>1</sup> 4*a*2 <sup>2</sup> + 2 *a*<sup>2</sup> 4*a*1 <sup>2</sup> + *a*<sup>2</sup> 6 ( *a*2 <sup>2</sup> + *a*<sup>1</sup> <sup>2</sup><sup>3</sup> *<sup>δ</sup>*<sup>1</sup> <sup>2</sup> <sup>−</sup> *a*2*b*<sup>1</sup> − *b*2*a*<sup>1</sup> + *a*<sup>1</sup> <sup>3</sup> + *a*1*a*<sup>2</sup> <sup>2</sup> *a*<sup>1</sup> <sup>3</sup> <sup>−</sup> *<sup>a</sup>*2*b*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*2*a*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*1*a*<sup>2</sup> 2 *δ*2 2) (70)

where *a*1, *δ*1, *δ*2, *a*2, *b*1, *b*<sup>2</sup> are some free constants and *a*<sup>1</sup> �= 0, *a*<sup>2</sup> �= 0. Now we can explicitly write down the three-wave solutions using

$$\begin{split} u(x,y,t) &= 2 \frac{2a\_1^{-2}\sqrt{\delta\_3}\cosh\left(\frac{\tilde{\varsigma}\_1 - \ln\sqrt{\delta\_3}}{2}\right) - \delta\_1 a\_2^{-2}\cos\tilde{\varsigma}\_2 + \delta\_2 a\_3^{-2}\cosh\tilde{\varsigma}\_3}{2\sqrt{\delta\_3}\cosh\left(\tilde{\varsigma}\_1 - \ln\sqrt{\delta\_3}\right) + \delta\_1 \cos\tilde{\varsigma}\_2 + \delta\_2 \cosh\tilde{\varsigma}\_3} \\ &- 2 \frac{\left(2\sqrt{\delta\_3}a\_1\sinh\left(\tilde{\varsigma}\_1 - \ln\sqrt{\delta\_3}\right) - \delta\_1 a\_2 \sin\tilde{\varsigma}\_2 + \delta\_2 a\_3 \sinh\tilde{\varsigma}\_3\right)^2}{\left(2\sqrt{\delta\_3}\cosh\left(\tilde{\varsigma}\_1 - \ln\sqrt{\delta\_3}\right) + \delta\_1 \cos\tilde{\varsigma}\_2 + \delta\_2 \cosh\tilde{\varsigma}\_3\right)^2} \end{split} \tag{71}$$

where *aj*, *bj*, *cj*, *δ<sup>j</sup>* (*j* = 1, 2, 3) satisfy Eq.(70). It is called the breather type multi-solitary waves solutions. Fig.8 is the plot of the spatial structure with the parameters selected as

$$
\begin{pmatrix} a\_1 \ b\_1 \ \delta\_1 \\ a\_2 \ b\_2 \ \delta\_2 \end{pmatrix} = \begin{pmatrix} 0.1 & 1 & 0.5 \\ 1.1 & 0.1 & 10^{-4} \end{pmatrix}
$$

Fig.8 is the process of interaction for two solitons solutions with the evolution of time, where *s*<sup>1</sup> and *s*<sup>2</sup> are the different two solitons, and *s*<sup>12</sup> represents the interaction between *s*1,*s*<sup>2</sup> and the periodic wave cos(*ξ*2). The phenomena of soliton interaction are clearly presented. It shows that the two soliton experience interaction, they will fusion with few oscillations and later travel ahead continuously. The value of *δ*<sup>2</sup> will determine the length of resonant soliton. Obviously, the solitons *s*<sup>1</sup> and *s*<sup>2</sup> can not approach each together closely from picture. They interact produce the breather wave.

Usually, the interactions between solitons for a lot of integrable or non-integrable system are considered to be completely elastic. That is to say, the amplitude, velocity and wave shape of a soliton do not change after the nonlinear collisions. However, for several nonlinear partial differential equations, completely nonelastic interactions will occur. On the other hand, two or more solitons will fusion to one soliton. These two types of phenomena was called soliton fission and soliton fusion, respectively. Now we demonstrate soliton fission and fusion of the Kadomtsev-Petviashvili equation with positive dispersion.

**Figure 8.** The plot of the space structure of the breather-type multi-solitary waves solutions and contour plot map of *u* in (*x*, *y*)-plane.

In Eq.(70), let *δ*<sup>3</sup> = 0, then

16 Will-be-set-by-IN-TECH

In the following, by using generalized three-wave type of Ans*atz* ¨ approach, we study the interaction and spatiotemporal feature of three-wave of KP equation with positive dispersion.

where *ξ*<sup>1</sup> = *a*1*x* + *b*1*y* + *c*1*t* + *θ*1, *ξ*<sup>2</sup> = *a*2*x* + *b*2*y* + *c*2*t* + *θ*2, *ξ*<sup>3</sup> = *a*3*x* + *b*3*y* + *c*3*t* + *θ*<sup>3</sup> and *aj*, *bj*, *cj*, *δj*, *j* = 1, 2, 3 are some constants to be determined. Substituting the Ans*atz* ¨ Eq.(69)

<sup>2</sup>*a*<sup>1</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*<sup>2</sup>

<sup>2</sup>*b*1*b*<sup>2</sup> + *a*<sup>2</sup>

<sup>2</sup> − *<sup>b</sup>*2*a*1) (*a*<sup>1</sup>

<sup>2</sup>*a*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

<sup>3</sup>*b*1*a*<sup>1</sup>

2*a*2

7*b*1

<sup>2</sup> + 2 *a*<sup>2</sup>

 − *δ*1*a*<sup>2</sup>

 0.1 1 0.5 1.1 0.1 10−<sup>4</sup>

<sup>2</sup> *a*<sup>1</sup>

<sup>4</sup>*a*<sup>1</sup> <sup>−</sup> <sup>6</sup> *<sup>a</sup>*2*b*1*b*<sup>2</sup>

<sup>2</sup>*a*1*a*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup>

<sup>3</sup> + *b*2*a*<sup>1</sup> + *a*1*a*<sup>2</sup>

3*a*2

<sup>4</sup>*b*<sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*<sup>1</sup>

<sup>6</sup>*b*<sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*1*a*<sup>2</sup>

<sup>3</sup> <sup>−</sup> *<sup>a</sup>*2*b*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*2*a*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*1*a*<sup>2</sup>

<sup>2</sup> cos *ξ*<sup>2</sup> + *δ*2*a*<sup>3</sup>

+ *δ*<sup>1</sup> cos *ξ*<sup>2</sup> + *δ*<sup>2</sup> cosh *ξ*<sup>3</sup>

− *δ*1*a*<sup>2</sup> sin *ξ*<sup>2</sup> + *δ*2*a*<sup>3</sup> sinh *ξ*<sup>3</sup>

+ *δ*<sup>1</sup> cos *ξ*<sup>2</sup> + *δ*<sup>2</sup> cosh *ξ*<sup>3</sup>

<sup>5</sup> <sup>−</sup> <sup>6</sup> *<sup>b</sup>*1*b*2*a*<sup>1</sup>

<sup>5</sup>*b*<sup>1</sup> + 3 *a*1*a*<sup>2</sup>

4*a*1 <sup>2</sup> + *a*<sup>2</sup> 6 

<sup>4</sup> + 9 *a*<sup>1</sup>

<sup>2</sup>*a*1*a*<sup>2</sup>

<sup>2</sup> − *<sup>a</sup>*2*b*1)

3*b*1 2*b*2

> 2*b*2 3

> > 2 *δ*2 2)

<sup>2</sup> cosh *ξ*<sup>3</sup>

2

2

(70)

(71)

<sup>5</sup>*a*<sup>1</sup> + *b*<sup>1</sup>

*<sup>ξ</sup>*<sup>1</sup> + *δ*<sup>1</sup> cos *ξ*<sup>2</sup> + *δ*<sup>2</sup> cosh *ξ*<sup>3</sup> + *δ*3*e*−*ξ*<sup>1</sup> (69)

**5.1. Fission and fusion of multi-wave**

Now we suppose the solution of Eq.(68) as

*<sup>a</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup> *<sup>b</sup>*2*a*<sup>1</sup> <sup>−</sup> *<sup>a</sup>*2*b*<sup>1</sup> *a*2 <sup>2</sup> + *a*<sup>1</sup>

<sup>5</sup> <sup>−</sup> <sup>2</sup> *<sup>a</sup>*<sup>1</sup>

<sup>2</sup> + *a*<sup>1</sup> 2) <sup>3</sup> (*a*<sup>1</sup>

<sup>4</sup>*a*<sup>2</sup> + 2 *a*<sup>1</sup>

<sup>3</sup> + 3 *a*<sup>1</sup>

2*b*1

<sup>7</sup> + 3 *a*2*b*1*a*<sup>1</sup>

3*a*2 <sup>3</sup> + *a*<sup>2</sup>

4 (*a*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>a</sup>*<sup>1</sup>

Δ<sup>1</sup> = 3 *b*2*a*<sup>1</sup>

+*a*<sup>1</sup> 3*b*2

Δ<sup>2</sup> = −

( *a*2 <sup>2</sup> + *a*<sup>1</sup>

+9 *a*1*a*<sup>2</sup>

 *b*2 2*a*1

*<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>a</sup>*<sup>1</sup>

*<sup>b</sup>*<sup>3</sup> <sup>=</sup> <sup>2</sup> *<sup>a</sup>*<sup>1</sup>

into Eq. (68) will produce the following relations:

3*a*2

*f* (*x*, *y*, *t*) = *e*

<sup>2</sup> , *<sup>c</sup>*<sup>3</sup> <sup>=</sup> <sup>Δ</sup><sup>1</sup> (*a*<sup>2</sup> <sup>2</sup> + *a*<sup>1</sup> 2) 3

<sup>2</sup>*a*<sup>1</sup> <sup>−</sup> <sup>3</sup> *<sup>b</sup>*<sup>1</sup>

<sup>5</sup>*a*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>1</sup>

(*a*<sup>2</sup> <sup>2</sup> + *a*<sup>1</sup> 2) 2

<sup>2</sup>*b*2*a*<sup>1</sup>

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 3*b*1

*a*2*b*<sup>1</sup> − *b*2*a*<sup>1</sup> + *a*<sup>1</sup>

<sup>2</sup> <sup>√</sup>*δ*<sup>3</sup> cosh

<sup>√</sup>*δ*<sup>3</sup> cosh

 *a*<sup>1</sup> *b*<sup>1</sup> *δ*<sup>1</sup> *a*<sup>2</sup> *b*<sup>2</sup> *δ*<sup>2</sup>

<sup>2</sup>*a*2*b*1*b*<sup>2</sup>

2*b*1 <sup>2</sup> + *a*<sup>1</sup> 4*a*2

<sup>2</sup>*a*<sup>2</sup> + 3 *b*<sup>1</sup>

<sup>5</sup> + 9 *a*<sup>2</sup>

<sup>2</sup> + 9 *a*<sup>1</sup>

<sup>3</sup> + 3 *a*<sup>2</sup>

<sup>3</sup> + *a*1*a*<sup>2</sup>

where *a*1, *δ*1, *δ*2, *a*2, *b*1, *b*<sup>2</sup> are some free constants and *a*<sup>1</sup> �= 0, *a*<sup>2</sup> �= 0. Now we can explicitly

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> ln <sup>√</sup>*δ*<sup>3</sup>

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> ln <sup>√</sup>*δ*<sup>3</sup>

solutions. Fig.8 is the plot of the spatial structure with the parameters selected as

 =

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> ln <sup>√</sup>*δ*<sup>3</sup>

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> ln <sup>√</sup>*δ*<sup>3</sup>

where *aj*, *bj*, *cj*, *δ<sup>j</sup>* (*j* = 1, 2, 3) satisfy Eq.(70). It is called the breather type multi-solitary waves

*a*2 <sup>2</sup> + *a*<sup>1</sup> 2

<sup>3</sup> + *a*2*b*<sup>1</sup> + *a*1*a*<sup>2</sup>

*a*2 <sup>2</sup> + *a*<sup>1</sup> 2

<sup>3</sup> <sup>−</sup> <sup>3</sup> *<sup>b</sup>*<sup>2</sup>

<sup>6</sup> + 9 *a*<sup>2</sup>

<sup>3</sup> <sup>−</sup> <sup>9</sup> *<sup>a</sup>*<sup>1</sup>

<sup>3</sup>*b*1*b*<sup>2</sup>

<sup>2</sup> <sup>−</sup> <sup>2</sup> *<sup>a</sup>*2*b*1*b*2*a*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

<sup>2</sup> <sup>−</sup>

2*a*<sup>1</sup>

 2 2

<sup>√</sup>*δ*3*a*<sup>1</sup> sinh

<sup>√</sup>*δ*<sup>3</sup> cosh

<sup>2</sup> + 3 *b*<sup>2</sup>

<sup>2</sup>*b*1*b*<sup>2</sup> + *a*<sup>1</sup>

*<sup>δ</sup>*<sup>3</sup> <sup>=</sup> <sup>Δ</sup><sup>2</sup>

2*a*2

<sup>2</sup>*a*2*b*<sup>1</sup>

<sup>2</sup>*b*<sup>2</sup> + 3 *a*<sup>2</sup>

<sup>2</sup><sup>3</sup> *<sup>δ</sup>*<sup>1</sup>

write down the three-wave solutions using

*u*(*x*, *y*, *t*) = 2

−2 2

$$\frac{\Delta\left( (a\_1^3 + a\_2b\_1 + a\_1a\_2^2 - a\_1b\_2)(a\_1^3 + a\_1b\_2 + a\_1a\_2^2 - a\_2b\_1)\delta\_2^{-2} - (a\_1^2 + a\_2^2)^3\delta\_1^{-2} \right)}{4(a\_1^2 + a\_2^2)^3(a\_1^3 + a\_1b\_2 + a\_1a\_2^2 - a\_2b\_1)(a\_1^3 + a\_2b\_1 + a\_1a\_2^2 - a\_1b\_2)} = 0$$

where Δ = *a*<sup>2</sup> 2 *a*2 <sup>2</sup> + *a*<sup>1</sup> <sup>2</sup><sup>2</sup> <sup>+</sup> (*a*2*b*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*2*a*1) 2 , *a*<sup>1</sup> �= 0 and *a*<sup>2</sup> �= 0.

For *a*<sup>2</sup> 2 *a*2 <sup>2</sup> + *a*<sup>1</sup> <sup>2</sup><sup>2</sup> <sup>+</sup> (*a*2*b*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*2*a*1) <sup>2</sup> can not be zero, then the condition must be satisfied:

$$(a\_1(a\_1^2 + a\_2^2 - b\_2) + a\_2b\_1)(a\_1(a\_1^2 + a\_2^2 + b\_2) - a\_2b\_1)\delta\_2^2 - (a\_1^2 + a\_2^2)^3\delta\_1^2 = 0$$

So, we obtain the following solution:

$$\begin{split} u(x,y,t) &= \frac{2(a\_1^2 e^{\xi\_1} - \delta\_1 a\_2^2 \cos \xi\_2 + \delta\_2 a\_3^2 \cosh \xi\_3)}{e^{\xi\_1} + \delta\_1 \cos \xi\_2 + \delta\_2 \cosh \xi\_3} \\ &- \frac{2\left(a\_1 e^{\xi\_1} - \delta\_1 a\_2 \sin \xi\_2 + \delta\_2 a\_3 \sinh \xi\_3\right)^2}{\left(e^{\xi\_1} + \delta\_1 \cos \xi\_2 + \delta\_2 \cosh \xi\_3\right)^2} \end{split} \tag{72}$$

#### 18 Will-be-set-by-IN-TECH 122 Wave Processes in Classical and New Solids Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations <sup>19</sup>

where *ξ*<sup>1</sup> = *a*1*x* + *b*1*y* + *c*1*t*, *ξ*<sup>2</sup> = *a*2*x* + *b*2*y* + *c*2*t*, *ξ*<sup>3</sup> = *a*3*x* + *b*3*y* + *c*3*t* and *aj*, *bj*, *cj*, *j* = 1, 2, 3, and *δ*1, *δ*<sup>2</sup> with the following relations:

together. This phenomena is called soliton fusion. However, in the right figure it is found that the breather wave with the period oscillation can split up into two smaller line solitons with

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations 123

Using Homoclinic test approach, Extend Homoclinic test approach, Three-wave method and Introducing parameters and small perturbation method, we obtain novel solutions of Potential Kadomtsev-Petviashvili equation and Kadomtsev-Petviashvili equation such as periodic solitary wave,breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave,kinky kink wave, periodic kink wave, two-solitary wave, doubly periodic wave, doubly breather solitary wave. Moreover, we observed that there were differently spatiotemporal features in two-wave and multi-wave propagations including the degeneracy of soliton, periodic bifurcation and soliton deflexion of two-wave, fission and fusion of breather two-wave and so on. In future, we intend to study the stability and the interactions patterns of *N*-wave solutions in KP equation. What's more, can we obtain similar results to another integrable or non-integrable system? How can one use the soliton fission and fusion of models to study the practically observed soliton fission and fusion in the

The work was supported by the National Natural Science Foundation of China (No. 10661028,

*College of Mathematics and Information Science, Qujing Normal University, Qujing 655000,*

*School of Information Sciences and Engineering, Yunnan University, Kunming 650091, P.R.China*

[1] Zabusky N, Kruskal M (1965) Interaction of "Solitons" in a Collisionless Plasma and the

[2] Ablowitz M, Clarkson P ( 1991) Soliton, nonlinear evolution equations and inverse

[3] Weiss J, Tabor M, Carnevale G (1983) The Painlevé property for partial differential

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*School of Mathematics and Physics, Yunnan University, Kunming 650091, P.R.China*

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different directions. This phenomena is called the soliton fission.

**6. Conclusion**

experiments?

**Acknowledgement**

10801037 and 11161055).

**Author details**

Jun Liu and Gui Mu

Zhengde Dai

*P.R.China*

Murong Jiang

**7. References**

$$\begin{aligned} c\_2 &= -\frac{3 \, b\_1^{\;1} a\_2 - a\_2^{\;5} - 3 \, b\_2^{\;2} a\_2 + 2 \, a\_1^{\;2} a\_2^{\;3} + 3 \, a\_1^{\;4} a\_2 - 6 \, a\_1 b\_1 b\_2}{a\_1^{\;2} + a\_2^{\;2}} \\\\ c\_1 &= -\frac{a\_1^{\;5} - 3 \, a\_2^{\;4} a\_1 - 6 \, a\_2 b\_1 b\_2 - 2 \, a\_1^{\;3} a\_2^{\;2} - 3 \, b\_1^{\;2} a\_1 + 3 \, b\_2^{\;2} a\_1}{a\_1^{\;2} + a\_2^{\;2}} \end{aligned}$$

$$\begin{array}{l} c\_{3} = (3\,b\_{2}a\_{1}^{7} + 3\,a\_{2}\,b\_{1}a\_{1}^{6} + 9\,a\_{2}^{2}\,b\_{2}a\_{1}^{5} + 9\,a\_{2}^{3}\,b\_{1}a\_{1}^{4} + 9\,a\_{1}^{3}\,a\_{2}^{4}\,b\_{2} + a\_{1}^{3}\,b\_{2}^{3} \\ c\_{1} = -3\,a\_{1}^{3}b\_{1}^{2}\,b\_{2} - 9\,a\_{1}^{2}a\_{2}\,b\_{1}b\_{2}^{2} + 9\,a\_{1}^{2}a\_{2}^{5}\,b\_{1} + 3\,a\_{1}^{2}\,b\_{1}^{3}a\_{2} + 3\,a\_{1}a\_{2}^{6}\,b\_{2} - 3\,a\_{1}a\_{2}^{2}\,b\_{2}^{3} \\ + 9\,a\_{1}a\_{2}^{2}\,b\_{1}^{2}\,b\_{2} + 3\,a\_{2}^{3}b\_{1}\,b\_{2}^{2} - b\_{1}^{3}a\_{2}^{3} + 3\,a\_{2}^{7}b\_{1})(\left(a\_{1}^{2} + a\_{2}^{2}\right)^{3})^{-1} \end{array}$$

$$\begin{aligned} a\_3 &= \frac{a\_2 b\_1 - b\_2 a\_1}{a\_1^2 + a\_2^2} \\ b\_3 &= \frac{a\_2 a\_1^{\frac{5}{2}} + 2 \, a\_2^{\frac{3}{2}} a\_1^{\frac{3}{2}} - a\_1^{-2} b\_1 b\_2 + a\_1 b\_1 \,^2 a\_2 + a\_1 a\_2^{\frac{2}{3}} - a\_1 b\_2 \,^2 a\_2 + b\_1 b\_2 a\_2^2}{\left(a\_1^2 + a\_2^2\right)^2} \\ \delta\_2^2 &= \frac{\left(a\_1^2 + a\_2^2\right)^3}{\left(a\_1^3 + a\_2 b\_1 + a\_1 a\_2^2 - b\_2 a\_1\right) \left(a\_1^3 + b\_2 a\_1 + a\_1 a\_2^2 - a\_2 b\_1\right)} \delta\_1^2 \end{aligned}$$

Fig.9 shows the plot of two kinds of interaction behavior between two single solitons with different parameters and a breather wave, where

$$
\begin{pmatrix} a\_1 \ b\_1 \ \delta\_1 \\ a\_2 \ b\_2 \ \delta\_2 \end{pmatrix} = \begin{pmatrix} -1.5 & 1 & 0.05 \\ -1.1 & 1.2 & 2 \end{pmatrix} \text{and} \begin{pmatrix} a\_1 \ b\_1 \ \delta\_1 \\ a\_2 \ b\_2 \ \delta\_2 \end{pmatrix} = \begin{pmatrix} 1.5 & -1 \ 0.05 \\ -1.1 & 1.2 & 2 \end{pmatrix}
$$

respectively. From the first picture of Fig.9, we can see that two single solitons interact strongly to make a resonance breather-wave solution from a point at which two incident solitons meet

**Figure 9.** Contourplot of the breather-type multi-solitary waves solutions with the different parameters.

together. This phenomena is called soliton fusion. However, in the right figure it is found that the breather wave with the period oscillation can split up into two smaller line solitons with different directions. This phenomena is called the soliton fission.
