**7. Rogue waves, "quantized" modulation instability, and dynamics of nonautonomous Peregrine solitons under "hyperbolic hurricane wind"**

Recently, a method of producing optical rogue waves, which are a physical counterpart to the rogue (monster) waves in oceans, have been developed (Solli et al., 2007). Optical rogue waves have been formed in the so-called soliton supercontinuum generation, a nonlinear optical process in which broadband "colored" solitons are generated from a narrowband optical background due to induced modulation instability and soliton fission effects (Dudley, 2009; Dudley et al., 2006; 2008).

Ordinary, the study of rogue waves has been focused on hydrodynamic applications and experiments (Clamond et al., 2006; Kharif & Pelinovsky, 2003)*.* Nonlinear phenomena in optical fibers also support rogue waves that are considered as soliton supercontinuum noise. It should be noticed that because optical rogue waves are closely related to oceanic rogue waves, the study of their properties opens novel possibilities to predict the dynamics of oceanic rogue waves. By using the mathematical equivalence between the propagation of nonlinear waves on water and the evolution of intense light pulses in optical fibers, an international research team (Kibler et al., 2010) recently reported the first observation of the so-called Peregrine soliton (Peregrine, 1983). Similar to giant nonlinear water waves, the Peregrine soliton solutions of the NLSE experience extremely rapid growth followed by just as rapid decay (Peregrine, 1983). Now, the Peregrine soliton is considered as a prototype of the famous ocean monster (rogue) waves responsible for many maritime catastrophes.

In this Section, the main attention will be focused on the possibilities of generation and amplification of nonautonomous Peregrine solitons. This study is an especially important for understanding how high intensity rogue waves may form in the very noisy and imperfect environment of the open ocean.

First of all, let us summarize the main features of the phenomenon known as the induced modulation instability. In 1984, Akira Hasegawa discovered that modulation instability of continuous (cw) wave optical signal in a glass fiber combined with an externally applied amplitude modulation can be utilized to produce a train of optical solitons (Hasegawa, 16 Will-be-set-by-IN-TECH

Fig. 3. Nonautonomous "agitated" breather (58) calculated within the framework of the model (46) after choosing the soliton management parameters *c*<sup>0</sup> = 0.25, *η*<sup>10</sup> = 0.5, *η*<sup>20</sup> = 1.5.

Recently, a method of producing optical rogue waves, which are a physical counterpart to the rogue (monster) waves in oceans, have been developed (Solli et al., 2007). Optical rogue waves have been formed in the so-called soliton supercontinuum generation, a nonlinear optical process in which broadband "colored" solitons are generated from a narrowband optical background due to induced modulation instability and soliton fission effects (Dudley, 2009;

Ordinary, the study of rogue waves has been focused on hydrodynamic applications and experiments (Clamond et al., 2006; Kharif & Pelinovsky, 2003)*.* Nonlinear phenomena in optical fibers also support rogue waves that are considered as soliton supercontinuum noise. It should be noticed that because optical rogue waves are closely related to oceanic rogue waves, the study of their properties opens novel possibilities to predict the dynamics of oceanic rogue waves. By using the mathematical equivalence between the propagation of nonlinear waves on water and the evolution of intense light pulses in optical fibers, an international research team (Kibler et al., 2010) recently reported the first observation of the so-called Peregrine soliton (Peregrine, 1983). Similar to giant nonlinear water waves, the Peregrine soliton solutions of the NLSE experience extremely rapid growth followed by just as rapid decay (Peregrine, 1983). Now, the Peregrine soliton is considered as a prototype of the famous

In this Section, the main attention will be focused on the possibilities of generation and amplification of nonautonomous Peregrine solitons. This study is an especially important for understanding how high intensity rogue waves may form in the very noisy and imperfect

First of all, let us summarize the main features of the phenomenon known as the induced modulation instability. In 1984, Akira Hasegawa discovered that modulation instability of continuous (cw) wave optical signal in a glass fiber combined with an externally applied amplitude modulation can be utilized to produce a train of optical solitons (Hasegawa,

**7. Rogue waves, "quantized" modulation instability, and dynamics of nonautonomous Peregrine solitons under "hyperbolic hurricane wind"**

ocean monster (rogue) waves responsible for many maritime catastrophes.

(a) the temporal behavior; (b) the corresponding contour map.

Dudley et al., 2006; 2008).

environment of the open ocean.

Fig. 4. Illustrative example of the temporal-spatial dynamics of the induced modulation instability and the Fermi-Pasta-Ulam recurrence effect calculated in the framework of the canonical NLSE model : (a) the intensity distribution; (b) the corresponding contour map.

1984). In the sense that the external modulation induces the modulation instability, Hasegawa called the total process as the induced modulation instability. To demonstrate the induced modulation instability (IMI), following Hasegawa, we solved the NLSE numerically with different depths and wavelength of modulation of cw wave. The main features of the induced modulation instability are presented in Fig.4. In Figure 4, following Hasegawa (Hasegawa, 1984), we present the total scenario of IMI and the restoration of the initial signal due to the Fermi-Pasta-Ulama recurrence effect. In our computer experiments, we have found novel and interesting feature of the IMI. Varying the depth of modulation and the level of continuous wave, we have discovered the effect which we called a "quantized" IMI. Figure 5 shows typical results of the computation. As can be clearly seen, the high-intensity IMI peaks are formed and split periodically into two, three, four, and more high-intensity peaks. In Fig.5 we present this splitting ("quantization") effect of the initially sinus like modulated cw signal into two and five high-intensity and "long-lived" components.

The Peregrine soliton can be considered as the utmost stage of the induced modulation instability, and its computer simulation is presented in Fig.6 When we compare the high-energy peaks of the IMI generated upon a distorted background (see Figs.4, 5) with exact form of the Peregrine soliton shown in Fig.7(a) we can understand, how such extreme wave structures may appear as they emerge suddenly on an irregular surface such as the open ocean.

There are two basic questions to be answered. What happens if arbitrary modulated cw wave is subjected to some form of external force? Such situations could include effects of wind, propagation of waves in nonuniform media with time dependent density gradients and slowly varying depth, nonlinearity and dispersion. For example, in Fig.7(b), we show the possibility of amplification of the Peregrine soliton when effects of wind are simulated by additional gain term in the canonical NLSE. The general questions naturally arise: To what extent the Peregrine soliton can be amplified under effects of wind, density gradients and

Fig. 6. Illustrative examples of the Peregrine soliton dynamics: (a) - classical Peregrine soliton calculated in the framework of the canonical NLSE model; (b) its behavior under linear

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 69

*<sup>R</sup>*<sup>3</sup> *<sup>X</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*<sup>κ</sup>*<sup>0</sup>*<sup>X</sup>* <sup>−</sup> <sup>2</sup>(*<sup>κ</sup>*<sup>0</sup>

The transformation (59) can be applied to obtain all solutions of the nonautonomous NLSE (30) and, in particular, the nonautonomous rational solutions known as the Peregrine solitons. Thus, the Peregrine soliton (Peregrine, 1983) can be discovered for the nonautonomous NLSE

*<sup>r</sup>*(*X*, *<sup>T</sup>*) = <sup>1</sup> <sup>−</sup> <sup>4</sup>(<sup>1</sup> <sup>+</sup> <sup>2</sup>*iT*)

*W*(*R*, *D*)

Figure 7 shows spatiotemporal behavior of the nonautonomous Peregrine soliton. The nonautonomous Peregrine soliton (63-65) shown in Fig.7(b) has been calculated in the framework of the nonautonomous NLSE model (28) after choosing the parameters *λ*<sup>0</sup> = Ω = 0, *D*<sup>2</sup> = *R*<sup>2</sup> = 1 and the gain coefficient Γ(*t*) = Γ0/(1 − Γ0*t*). Somewhat surprisingly, however, this figure indicates a sharp compression and strong amplification of the nonautonomous Peregrine soliton under the action of hyperbolic gain which, in particular, in the open ocean

It should be stressed that since the nonautonomous NLSE model is applied in many other physical systems such as plasmas and Bose-Einstein condensates (BEC), the results obtained in this Section can stimulate new research directions in many novel fields (see, for example,

2

<sup>2</sup> <sup>−</sup> *<sup>η</sup>*<sup>0</sup>

*qP*(*x*, *t*) = *A*(*t*)*r*(*X*, *T*) exp [*iφ*(*T*)] (63)

*κ*0; *P*(*t*) = *R*(*t*)/*D*(*t*). (62)

<sup>2</sup>)*T*(*t*)

<sup>1</sup> <sup>+</sup> <sup>4</sup>*T*<sup>2</sup> <sup>+</sup> <sup>4</sup>*X*<sup>2</sup> , (64)

*<sup>R</sup>*<sup>3</sup> *<sup>X</sup>*<sup>2</sup> <sup>+</sup> *<sup>T</sup>*(*t*) (65)

,

*W*(*R*, *D*)

*R*0

*<sup>φ</sup>*(*X*, *<sup>T</sup>*) = <sup>1</sup>

amplification associated with continuous wind.

*<sup>η</sup>*<sup>0</sup> <sup>=</sup> *<sup>D</sup>*<sup>0</sup> *R*0

model as well

where

<sup>×</sup> exp *i* 1 2

can be associated with "hyperbolic hurricane wind".

(Bludov et al., 2009; Yan, 2010)).

*<sup>η</sup>*0; *<sup>κ</sup>*<sup>0</sup> <sup>=</sup> *<sup>D</sup>*<sup>0</sup>

Fig. 5. Illustrative example of the "quantized" induced modulation instability: (a) the temporal-spatial behavior; (b) the corresponding contour map.

slowly varying depth, nonlinearity and dispersion? To answer these questions, let us consider the dynamics of the Peregrine soliton in the framework of the nonautonomous NLSE model. In the previous chapters, the auto -Bäcklund transformation has been used to find soliton solutions of the nonautonomous NLSE model. Now, we consider another remarkable method to study nonautonomous solitons. The following transformation

$$q(\mathbf{x}, t) = A(t)u(\mathbf{X}, T) \exp\left[i\phi(\mathbf{X}, T)\right] \tag{59}$$

has been used by Serkin and Hasegawa in (Serkin & Hasegawa, 2000a;b; 2002) to reduce the nonautonomous NLSE with varying dispersion, nonlinearity and gain or loss to the "ideal" NLSE

$$i\frac{\partial u}{\partial T} + \frac{\sigma}{2} \frac{\partial^2 u}{\partial X^2} + |u|^2 u = 0,$$

where the following notations may be introduced

$$A(t) = \sqrt{P(t)}; \quad \mathbf{X} = P(t)\mathbf{x}; \quad T(t) = \int\_0^t D(\tau)P^2(\tau)d\tau;\tag{60}$$

$$\phi(\mathbf{X},T) = \frac{1}{2}\frac{\mathcal{W}(\mathbf{R},D)}{R^3}\mathbf{X}^2 - \varrho\left(\mathbf{X},T\right),\tag{61}$$

where *ϕ* (*X*, *T*) is the phase of the canonical soliton.

It is easy to see that by using Eq.(59-61), the one-soliton solution may be written in the following form

$$q\_1^+(\mathbf{x}, t \mid \sigma = +1) = 2\tilde{\eta\_0}A(t)\text{sech}\left[2\tilde{\eta\_0}X + 4\tilde{\eta\_0}\tilde{\kappa\_0}T(t)\right],$$

Fig. 6. Illustrative examples of the Peregrine soliton dynamics: (a) - classical Peregrine soliton calculated in the framework of the canonical NLSE model; (b) its behavior under linear amplification associated with continuous wind.

$$\times \exp\left\{i\left[\frac{1}{2}\frac{W(\mathbb{R},D)}{R^3}\mathbf{X}^2 - 2\tilde{\kappa}\_0\mathbf{X} - 2(\tilde{\kappa}\_0^{-2} - \tilde{\eta}\_0^{-2})T(t)\right]\right\},$$

$$\eta\tilde{\eta}\_0 = \frac{D\_0}{R\_0}\eta\_0;\ \tilde{\kappa}\_0 = \frac{D\_0}{R\_0}\kappa\_0;\ \mathcal{P}(t) = \mathcal{R}(t)/D(t). \tag{62}$$

The transformation (59) can be applied to obtain all solutions of the nonautonomous NLSE (30) and, in particular, the nonautonomous rational solutions known as the Peregrine solitons. Thus, the Peregrine soliton (Peregrine, 1983) can be discovered for the nonautonomous NLSE model as well

$$q\_P(\mathbf{x}, t) = A(t)r(\mathbf{X}, T) \exp\left[i\phi(T)\right] \tag{63}$$

where

18 Will-be-set-by-IN-TECH

Fig. 5. Illustrative example of the "quantized" induced modulation instability: (a) the

slowly varying depth, nonlinearity and dispersion? To answer these questions, let us consider the dynamics of the Peregrine soliton in the framework of the nonautonomous NLSE model. In the previous chapters, the auto -Bäcklund transformation has been used to find soliton solutions of the nonautonomous NLSE model. Now, we consider another remarkable method

has been used by Serkin and Hasegawa in (Serkin & Hasegawa, 2000a;b; 2002) to reduce the nonautonomous NLSE with varying dispersion, nonlinearity and gain or loss to the "ideal"

*P*(*t*); *X* = *P*(*t*)*x*; *T*(*t*) =

*W*(*R*, *D*)

It is easy to see that by using Eq.(59-61), the one-soliton solution may be written in the

<sup>1</sup> (*x*, *<sup>t</sup>* <sup>|</sup> *<sup>σ</sup>* = +1) = <sup>2</sup>*<sup>η</sup>*<sup>0</sup>*A*(*t*)sech [2*<sup>η</sup>*<sup>0</sup>*<sup>X</sup>* <sup>+</sup> <sup>4</sup>*<sup>η</sup>*<sup>0</sup>*<sup>κ</sup>*<sup>0</sup>*T*(*t*)]

2

*q*(*x*, *t*) = *A*(*t*)*u*(*X*, *T*) exp [*iφ*(*X*, *T*)] (59)

<sup>2</sup> *u* = 0,

 *t*

*D*(*τ*)*P*2(*τ*)*dτ*; (60)

*<sup>R</sup>*<sup>3</sup> *<sup>X</sup>*<sup>2</sup> <sup>−</sup> *<sup>ϕ</sup>* (*X*, *<sup>T</sup>*), (61)

0

temporal-spatial behavior; (b) the corresponding contour map.

to study nonautonomous solitons. The following transformation

*i ∂u <sup>∂</sup><sup>T</sup>* <sup>+</sup> *σ* 2 *∂*2*u <sup>∂</sup>X*<sup>2</sup> <sup>+</sup> <sup>|</sup>*u*<sup>|</sup>

*<sup>φ</sup>*(*X*, *<sup>T</sup>*) = <sup>1</sup>

where the following notations may be introduced

where *ϕ* (*X*, *T*) is the phase of the canonical soliton.

*A*(*t*) =

*q*+

NLSE

following form

$$r(X,T) = 1 - \frac{4(1+2iT)}{1+4T^2+4X^2} \tag{64}$$

$$
\phi(X, T) = \frac{1}{2} \frac{W(R, D)}{R^3} X^2 + T(t) \tag{65}
$$

Figure 7 shows spatiotemporal behavior of the nonautonomous Peregrine soliton. The nonautonomous Peregrine soliton (63-65) shown in Fig.7(b) has been calculated in the framework of the nonautonomous NLSE model (28) after choosing the parameters *λ*<sup>0</sup> = Ω = 0, *D*<sup>2</sup> = *R*<sup>2</sup> = 1 and the gain coefficient Γ(*t*) = Γ0/(1 − Γ0*t*). Somewhat surprisingly, however, this figure indicates a sharp compression and strong amplification of the nonautonomous Peregrine soliton under the action of hyperbolic gain which, in particular, in the open ocean can be associated with "hyperbolic hurricane wind".

It should be stressed that since the nonautonomous NLSE model is applied in many other physical systems such as plasmas and Bose-Einstein condensates (BEC), the results obtained in this Section can stimulate new research directions in many novel fields (see, for example, (Bludov et al., 2009; Yan, 2010)).

Fig. 8. Nonautonomous KdV solitons calculated within the framework of the model (71) after choosing the soliton management parameters *α* = 0.15, *η*<sup>10</sup> = 0.40, *η*<sup>20</sup> = 0.75. On the left hand side the temporal behavior is presented, while the corresponding contour map is

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 71

Let us consider the simplest option to choose the real solution *Q*(*x*, *t*), which leads to the only possibility of *ϕ* = *λ*<sup>1</sup> = 0. In this case, Eq.(67) is reduced to the KdV with variable coefficients

where the notation *R*3(*t*) = *F*2*γD*3(*t*) has been introduced. It is easy to verify that Eq.(68) can

*R*3(*T*)

*qt* − 6*σqqx* + *qxxx* = 0. Applying the auto-Backlund transformation, we can write down the two-soliton solution of

<sup>2</sup>*β*<sup>1</sup> sinh *<sup>ξ</sup>*<sup>1</sup> sinh *<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*β*<sup>2</sup> cosh *<sup>ξ</sup>*<sup>1</sup> cosh *<sup>ξ</sup>*<sup>2</sup>

*<sup>Q</sup>*(*x*, *<sup>t</sup>*) = *<sup>D</sup>*3(*T*)

*D*3(*τ*)*dτ* so that *q*(*x*, *T*) is given by the canonical KdV:

*Q*2(*x*, *t*) = −2*σ*(*β*<sup>1</sup> − *β*2)

N<sup>1</sup> = *β*<sup>1</sup> (sinh *ξ*2)

*<sup>ξ</sup><sup>i</sup>* <sup>=</sup> *<sup>β</sup>i*/2 (*<sup>x</sup>* <sup>−</sup> <sup>2</sup>*βiT*), *<sup>β</sup><sup>i</sup>* <sup>=</sup> <sup>2</sup>*η*<sup>2</sup>

1 2

*q*(*x*, *T*),

*D*3(*T*) *R*3(*T*)

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

0*i*

, *i* = 1, 2;

N<sup>1</sup> D<sup>1</sup>

*W*(*D*3, *R*3) *D*3*R*<sup>3</sup>

= 0, (68)

, (69)

<sup>2</sup> , (70)

2 ,

*Qt* − 6*σR*3(*t*)*QQx* + *D*3(*t*)*Qxxx* +

be mapped into the standard KdV under the transformations

presented on the right hand side.

where *T* =

where

*t* 0

the nonautonomous KdV

D<sup>1</sup> =

and *η*<sup>02</sup> > *η*<sup>01</sup> are initial amplitudes of the solitons.

Fig. 7. (a) Autonomous and (b) nonautonomous Peregrine solitons calculated within the framework of the model (63-65) after choosing the soliton management parameters Γ<sup>0</sup> = 0.33.

### **8. Nonautonomous KdV solitons**

Notice, that the nonlinear evolution equations that arise in the approach of variable spectral parameter contain, as a rule, an explicit dependence on the coordinates. Our general approach makes it possible to construct not only the well-known equations, but also a number of new integrable equations (NLSE, KdV, modified KdV, Hirota and Satsuma and so on) by extending the Zakharov–Shabat (ZS) and AKNS formalism. In particular, Eqs.(9,10) under the conditions (11) with *a*2=0, *a*3=−4*iD*<sup>3</sup> and *R*=1 become

$$Q\_T = -D\_5 Q\_{SSS} S\_x^3 - 6iD\_3 Q\_{SS} \rho\_S S\_x^3 + 3i D\_3 \sigma F^{2\gamma} Q^2 \rho\_S S\_x + 6D\_3 \sigma F^{2\gamma} Q Q\_S S\_x \tag{66}$$

$$+ Q\_S \left( -S\_t + \lambda\_1 S - V\_1 S\_x - 6iD\_3 \rho\_{SS} S\_x^3 + \frac{3}{4} D\_3 \rho\_S^2 S\_x^3 \right)$$

$$-iQ \left[ 2\lambda\_0 S / S\_x - 2\gamma + \frac{1}{2} \left( \rho\_T + \rho\_S S\_t \right) - \frac{1}{2} \lambda\_1 S \rho\_S + \frac{1}{2} V \rho\_S S\_x \right]$$

$$+ Q \left( \lambda\_1 - \gamma \frac{F\_T}{F} + \frac{3}{4} D\_3 \rho\_S \rho\_{SS} S\_x^3 \right) - iQ \left( -\frac{1}{8} D\_3 \rho\_S^3 S\_x^3 + \frac{1}{2} D\_3 \rho\_{SSS} S\_x^3 \right),$$

Eq.(66) can be rewritten in the independent variables (*x*, *t*)

$$\begin{split} Q\_t &= -D\_3 Q\_{\text{xxx}} - 6i D\_3 Q\_{\text{xx}} \varrho\_x + 3i D\_3 \sigma F^{2\gamma} Q^2 \varrho\_x + 6D\_3 \sigma F^{2\gamma} Q Q\_x \\ &+ Q\_x \left( \lambda\_1 S / S\_x - V\_1 - 6i D\_3 \varrho\_{\text{xx}} + \frac{3}{4} D\_3 \varrho\_x^2 \right) \\ &- iQ \left[ 2\lambda\_0 S / S\_x - 2\gamma + \frac{1}{2} \left( \varrho\_T + \varrho\_S S\_t \right) - \frac{1}{2} \lambda\_1 S \varrho\_x / S\_x + \frac{1}{2} V \varrho\_x \right] \\ &+ Q \left( \lambda\_1 - \gamma \frac{F\_t}{F} + \frac{3}{4} D\_3 \varrho\_X \varrho\_{\text{xx}} \right) - iQ \left( -\frac{1}{8} D\_3 \varrho\_x^3 + \frac{1}{2} D\_3 \varrho\_{\text{xxx}} \right) . \end{split}$$

Fig. 8. Nonautonomous KdV solitons calculated within the framework of the model (71) after choosing the soliton management parameters *α* = 0.15, *η*<sup>10</sup> = 0.40, *η*<sup>20</sup> = 0.75. On the left hand side the temporal behavior is presented, while the corresponding contour map is presented on the right hand side.

Let us consider the simplest option to choose the real solution *Q*(*x*, *t*), which leads to the only possibility of *ϕ* = *λ*<sup>1</sup> = 0. In this case, Eq.(67) is reduced to the KdV with variable coefficients

$$Q\_l - 6\sigma R\_3(t) Q Q\_x + D\_3(t) Q\_{\text{xxx}} + \frac{1}{2} \frac{W(D\_3, R\_3)}{D\_3 R\_3} = 0,\tag{68}$$

where the notation *R*3(*t*) = *F*2*γD*3(*t*) has been introduced. It is easy to verify that Eq.(68) can be mapped into the standard KdV under the transformations

$$Q(\mathbf{x}, t) = \frac{D\_3(T)}{R\_3(T)} q(\mathbf{x}, T) \rho$$

where *T* = *t* 0 *D*3(*τ*)*dτ* so that *q*(*x*, *T*) is given by the canonical KdV:

$$q\_t - \theta \sigma q q\_{\ge} + q\_{\text{xxx}} = 0.$$

Applying the auto-Backlund transformation, we can write down the two-soliton solution of the nonautonomous KdV

$$Q\_2(\mathbf{x}, t) = -2\sigma(\beta\_1 - \beta\_2) \frac{D\_3(T)}{R\_3(T)} \frac{\mathfrak{N}\_1}{\mathfrak{D}\_1},\tag{69}$$

where

20 Will-be-set-by-IN-TECH

Fig. 7. (a) Autonomous and (b) nonautonomous Peregrine solitons calculated within the framework of the model (63-65) after choosing the soliton management parameters Γ<sup>0</sup> = 0.33.

Notice, that the nonlinear evolution equations that arise in the approach of variable spectral parameter contain, as a rule, an explicit dependence on the coordinates. Our general approach makes it possible to construct not only the well-known equations, but also a number of new integrable equations (NLSE, KdV, modified KdV, Hirota and Satsuma and so on) by extending the Zakharov–Shabat (ZS) and AKNS formalism. In particular, Eqs.(9,10) under the conditions

<sup>2</sup> (*ϕ<sup>T</sup>* <sup>+</sup> *<sup>ϕ</sup>SSt*) <sup>−</sup> <sup>1</sup>

*x* − *iQ* −1 8 *D*3*ϕ*<sup>3</sup> *SS*3 *<sup>x</sup>* + 1 2

*<sup>x</sup>* + 3 4 *D*3*ϕ*<sup>2</sup> *SS*3 *x* 

2

*Qt* <sup>=</sup> <sup>−</sup>*D*3*Qxxx* <sup>−</sup> <sup>6</sup>*iD*3*Qxxϕ<sup>x</sup>* <sup>+</sup> <sup>3</sup>*iD*3*σF*2*γQ*2*ϕ<sup>x</sup>* <sup>+</sup> <sup>6</sup>*D*3*σF*2*γQQx* (67)

3 4 *D*3*ϕ*<sup>2</sup> *x* 

<sup>2</sup> (*ϕ<sup>T</sup>* <sup>+</sup> *<sup>ϕ</sup>SSt*) <sup>−</sup> <sup>1</sup>

 − *iQ* −1 8 *D*3*ϕ*<sup>3</sup> *<sup>x</sup>* + 1 2 *D*3*ϕxxx*

*λ*1*Sϕ<sup>S</sup>* +

2

*<sup>x</sup>* + <sup>3</sup>*iD*3*σF*2*γQ*2*ϕSSx* + <sup>6</sup>*D*3*σF*2*<sup>γ</sup>QQSSx* (66)

1 2 *VϕSSx* 

*λ*1*Sϕx*/*Sx* +

*D*3*ϕSSSS*<sup>3</sup> *x* ,

> 1 2 *Vϕx*

> > .

**8. Nonautonomous KdV solitons**

(11) with *a*2=0, *a*3=−4*iD*<sup>3</sup> and *R*=1 become

*<sup>x</sup>* <sup>−</sup> <sup>6</sup>*iD*3*QSSϕSS*<sup>3</sup>

2*λ*0*S*/*Sx* − 2*γ* +

*FT <sup>F</sup>* <sup>+</sup> 3 4

Eq.(66) can be rewritten in the independent variables (*x*, *t*)

*λ*<sup>1</sup> − *γ*

2*λ*0*S*/*Sx* − 2*γ* +

*Ft <sup>F</sup>* <sup>+</sup> 3 4

*λ*<sup>1</sup> − *γ*

+*Qx* 

−*iQ* 

+*Q* 

<sup>−</sup>*St* <sup>+</sup> *<sup>λ</sup>*1*<sup>S</sup>* <sup>−</sup> *<sup>V</sup>*1*Sx* <sup>−</sup> <sup>6</sup>*iD*3*ϕSSS*<sup>3</sup>

1

*λ*1*S*/*Sx* − *V*<sup>1</sup> − 6*iD*3*ϕxx* +

1

*D*3*ϕxϕxx*

*D*3*ϕSϕSSS*<sup>3</sup>

*QT* <sup>=</sup> <sup>−</sup>*D*3*QSSSS*<sup>3</sup>

+*QS* 

−*iQ* 

+*Q* 

$$\mathcal{W}\_1 = \beta\_1 \left(\sinh \tilde{\varrho}\_2\right)^2 + \beta\_2 \left(\cosh \tilde{\varrho}\_1\right)^2,\tag{70}$$

$$\begin{aligned} \mathfrak{D}\_1 &= \left[ \sqrt{2\beta\_1} \sinh \xi\_1 \sinh \xi\_2 - \sqrt{2\beta\_2} \cosh \xi\_1 \cosh \xi\_2 \right]^2, \\ \mathfrak{F}\_i &= \sqrt{\beta\_i/2} \left( \mathbf{x} - 2\beta\_i T \right), \ \beta\_i = 2\eta\_{0i'}^2 \ \mathbf{i} = 1, 2; \end{aligned}$$

and *η*<sup>02</sup> > *η*<sup>01</sup> are initial amplitudes of the solitons.

the inverse scattering transform method. We have derived the laws of a soliton adaptation to the external potential. It is precisely this soliton adaptation mechanism which was of prime physical interest in our Chapter. We clarified some examples in order to gain a better understanding into this physical mechanism which can be considered as the interplay between nontrivial time-dependent parabolic soliton phase and external time-dependent potential. We stress that this nontrivial time-space dependent phase profile of nonautonomous soliton depends on the Wronskian of nonlinearity *R*(*t*) and dispersion *D*(*t*) and this profile

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 73

Several novel analytical solutions for water waves have been presented. In particular, we have found novel solutions for the generalized Peregrine solitons in inhomogeneous and nonautonomous systems, "quantized" modulation instability, and the exactly integrable model for the Peregrine solitons under "hyperbolic hurricane wind". It was shown that important mathematical analogies between optical rogue waves and the Peregrine solitons in water open the possibility to study optical rogue waves and water rogue waves in parallel and, due to the evident complexity of experiments with rogue waves in oceans, this method offers remarkable possibilities in studies nonlinear hydrodynamics problems by performing

We would like to conclude by saying that the concept of adaptation is of primary importance in nature and nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the

This investigation is a natural follow up of the works performed in collaboration with Professor Akira Hasegawa and the authors would like to thank him for this collaboration.

Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. (1973). Nonlinear-evolution equations

Akhmediev, N. N. & Ankiewicz, A. (1997). *Solitons. Nonlinear pulses and beams*, Charman and

Akhmediev, N. N. & Ankiewicz, A. (2008). *Dissipative Solitons: From Optics to Biology and*

Atre, R., Panigrahi, P. K. & Agarwal, G. S. (2006). Class of solitary wave solutions of the one-dimensional gross-pitaevskii equation, *Phys. Rev. E* 73(5): 056611. Avelar, A. T., Bazeia, D. & Cardoso, W. B. (2009). Solitons with cubic and quintic nonlinearities

Balakrishnan, R. (1985). Soliton propagation in nonuniform media, *Phys. Rev. A*

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does not exist for canonical NLSE soliton when *R*(*t*) = *D*(*t*) = 1.

experiments in the nonlinear optical systems.

**10. References**

Hall, London.

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Fig. 9. Nonautonomous KdV solitons calculated within the framework of the model (72) after choosing the soliton management parameters *α* = 2.0, *β* = −0.25, *η*<sup>10</sup> = 0.40, *η*<sup>20</sup> = 0.75.

As two illustrative examples, in Fig.8, we present the behavior of nonautonomous KdV soliton in the framework of the model

$$Q\_t - 6\sigma Q Q\_x + \exp(\mathfrak{a}t)Q\_{\text{xxx}} - \frac{1}{2}\mathfrak{a}Q = 0\tag{71}$$

with lineal gain (or loss) accompanying by exponential variation of the dispersion coefficient; and in Fig.9 we show the dynamics of the KdV soliton in the nonautonomous system described by the model

$$\left[Q\_l - \delta \sigma Q Q\_x + \left[1 + \beta \cos(\alpha t)\right]/(1+\beta)Q\_{\text{xxx}} + \frac{a\beta \sin(\alpha t)}{2\left[1 + \beta \cos(\alpha t)\right]/(1+\beta)}Q = 0\tag{72}$$

where *D*3(*t*) = [1 + *β* cos(*αt*)] /(1 + *β*), *R*3(*t*) = 1.

It is important to compare our exactly integrable nonautonomous KdV model with the model proposed by Johnson to describe the KdV soliton dynamics under the influence of the depth variation (Johnson, 1997) and given by

$$
\mu\_X - 6\sigma \mathcal{D}(X)^{-3/2} u \mu\_\xi + \mathcal{D}(X)^{1/2} u\_{\tilde{\xi}\tilde{\xi}\tilde{\xi}} + \frac{1}{2} \frac{\mathcal{D}\_X}{\mathcal{D}} u = 0. \tag{73}
$$

We stress that after choosing the parameters *<sup>R</sup>*3(*t*) = <sup>D</sup>(*t*)−3/2 and *<sup>D</sup>*3(*t*) = <sup>D</sup>(*t*)1/2, the potential in Eq.(68) becomes *<sup>W</sup>*(*D*3,*R*3) *<sup>D</sup>*3*R*<sup>3</sup> = −2D� /D, which is very nearly similar to the potential in Eq.(73) calculated by Johnson (Johnson, 1997).

#### **9. Conclusions**

The solution technique based on the generalized Lax pair operator method opens the possibility to study in details the nonlinear dynamics of solitons in nonautonomous nonlinear and dispersive physical systems. We have focused on the situation in which the generalized nonautonomous NLSE model was found to be exactly integrable from the point of view of the inverse scattering transform method. We have derived the laws of a soliton adaptation to the external potential. It is precisely this soliton adaptation mechanism which was of prime physical interest in our Chapter. We clarified some examples in order to gain a better understanding into this physical mechanism which can be considered as the interplay between nontrivial time-dependent parabolic soliton phase and external time-dependent potential. We stress that this nontrivial time-space dependent phase profile of nonautonomous soliton depends on the Wronskian of nonlinearity *R*(*t*) and dispersion *D*(*t*) and this profile does not exist for canonical NLSE soliton when *R*(*t*) = *D*(*t*) = 1.

Several novel analytical solutions for water waves have been presented. In particular, we have found novel solutions for the generalized Peregrine solitons in inhomogeneous and nonautonomous systems, "quantized" modulation instability, and the exactly integrable model for the Peregrine solitons under "hyperbolic hurricane wind". It was shown that important mathematical analogies between optical rogue waves and the Peregrine solitons in water open the possibility to study optical rogue waves and water rogue waves in parallel and, due to the evident complexity of experiments with rogue waves in oceans, this method offers remarkable possibilities in studies nonlinear hydrodynamics problems by performing experiments in the nonlinear optical systems.

We would like to conclude by saying that the concept of adaptation is of primary importance in nature and nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity changes can be fundamental objects of nonlinear science.

This investigation is a natural follow up of the works performed in collaboration with Professor Akira Hasegawa and the authors would like to thank him for this collaboration. We thank BUAP and CONACyT, Mexico for support.

#### **10. References**

22 Will-be-set-by-IN-TECH

Fig. 9. Nonautonomous KdV solitons calculated within the framework of the model (72) after choosing the soliton management parameters *α* = 2.0, *β* = −0.25, *η*<sup>10</sup> = 0.40, *η*<sup>20</sup> = 0.75.

As two illustrative examples, in Fig.8, we present the behavior of nonautonomous KdV soliton

with lineal gain (or loss) accompanying by exponential variation of the dispersion coefficient; and in Fig.9 we show the dynamics of the KdV soliton in the nonautonomous system

It is important to compare our exactly integrable nonautonomous KdV model with the model proposed by Johnson to describe the KdV soliton dynamics under the influence of the depth

We stress that after choosing the parameters *<sup>R</sup>*3(*t*) = <sup>D</sup>(*t*)−3/2 and *<sup>D</sup>*3(*t*) = <sup>D</sup>(*t*)1/2, the

The solution technique based on the generalized Lax pair operator method opens the possibility to study in details the nonlinear dynamics of solitons in nonautonomous nonlinear and dispersive physical systems. We have focused on the situation in which the generalized nonautonomous NLSE model was found to be exactly integrable from the point of view of

2

2 [1 + *β* cos(*αt*)](1 + *β*)

/D, which is very nearly similar to the potential

1 2 D*X*

*αQ* = 0 (71)

<sup>D</sup> *<sup>u</sup>* <sup>=</sup> 0. (73)

*Q* = 0 (72)

*Qt* <sup>−</sup> <sup>6</sup>*σQQx* <sup>+</sup> exp(*αt*)*Qxxx* <sup>−</sup> <sup>1</sup>

*Qt* <sup>−</sup> <sup>6</sup>*σQQx* <sup>+</sup> [<sup>1</sup> <sup>+</sup> *<sup>β</sup>* cos(*αt*)] /(<sup>1</sup> <sup>+</sup> *<sup>β</sup>*)*Qxxx* <sup>+</sup> *αβ* sin(*αt*)

*uX* <sup>−</sup> <sup>6</sup>*σ*D(*X*)−3/2*uu<sup>ξ</sup>* <sup>+</sup> <sup>D</sup>(*X*)1/2*uξξξ* <sup>+</sup>

*<sup>D</sup>*3*R*<sup>3</sup> = −2D�

in the framework of the model

where *D*3(*t*) = [1 + *β* cos(*αt*)] /(1 + *β*), *R*3(*t*) = 1.

in Eq.(73) calculated by Johnson (Johnson, 1997).

variation (Johnson, 1997) and given by

potential in Eq.(68) becomes *<sup>W</sup>*(*D*3,*R*3)

**9. Conclusions**

described by the model


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**4** 

*Russia* 

**Planar Stokes Flows with Free Boundary** 

The quasi-stationary Stokes approximation (Frenkel, 1945; Happel & Brenner, 1965) is used to describe viscous flows with small Reynolds numbers. Two-dimensional Stokes flow with free boundary attracted the attention of many researches. In particular, an analogy is drawn (Ionesku, 1965) between the equations of the theory of elasticity (Muskeleshvili, 1966) and the equations of hydrodynamics in the Stokes approximation. This idea allowed (Antanovskii, 1988) to study the relaxation of a simply connected cylinder under the effect of capillary forces. Hopper (1984) proposed to describe the dynamics of the free boundary through a family of conformal mappings. This approach was later used in (Jeong & Moffatt, 1992; Tanveer & Vasconcelos, 1994) for analysis of

We have developed a method of flow calculation, which is based on the expansion of pressure in a complete system of harmonic functions. The structure of this system depends on the topology of the region. Using the pressure distribution, we calculate the velocity on the boundary and investigate the motion of the boundary. In case of capillary forces the pressure is the projection of a generalized function with the carrier on the boundary on the

We show that in the 2D case there exists a non-trivial variation of pressure and velocity which keeps the Reynolds stress tensor unchanged. The correspondent variations of pressure give us the basis for pressure presentation in form of a series. Using this fact and the variation formulation of the Stokes problem we obtain a system of equations for the coefficients of this series. The variations of velocity give us the basis for the vortical part of velocity presentation in the form of a serial expansion with the same coefficients as for the

We obtain the potential part of velocity on the boundary directly from the boundary conditions - known external stress applied to the boundary. After calculating velocity on the boundary with given shape we calculate the boundary deformation during a small time

Based on this theory we have developed a method for calculation of the planar Stokes flows driven by arbitrary surface forces and potential volume forces. We can apply this method for investigating boundary deformation due to capillary forces, external pressure,

**1. Introduction** 

pressure series.

centrifugal forces, etc.

step.

free-surface cusps and bubble breakup.

subspace of harmonic functions (Chivilikhin, 1992).

Sergey Chivilikhin1 and Alexey Amosov2 *1National Research University of Information* 

*2Corning Scientific Center, Corning Incorporated* 

*Technologies, Mechanics and Optics,* 

