**3. Generalized nonlinear Schrödinger equation and solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons**

Let us study a special case of the reduction procedure for Eqs. (9,10) when *a*<sup>3</sup> = 0

$$\begin{split} A &= -i\lambda\_0 \mathcal{S}/\mathcal{S}\_X + a\_0(T) - \frac{1}{2} a\_2(T) \sigma F^{2\gamma} \left| Q \right|^2 - i\lambda\_1 \mathcal{S}/\mathcal{S}\_X \Lambda + a\_1(T) \Lambda + a\_2(T) \Lambda^2 \\\\ B &= \sqrt{\sigma} F^{\gamma} \exp \left( iq/2 \right) \left\{ -\frac{i}{4} a\_2(T) Q q \rho\_S S\_X - \frac{1}{2} a\_2(T) Q\_S S\_X \right\} + \\\\ i \left\{ Q \left[ -i\lambda\_1 \mathcal{S}/\mathcal{S}\_X + a\_1(T) + \Lambda a\_2(T) \right] \right\} \\\\ \mathcal{C} &= \sqrt{\sigma} F^{\gamma} \exp \left( -iq/2 \right) \left\{ \frac{i}{4} a\_2(T) Q^\* \rho\_S S\_X - \frac{1}{2} a\_2(T) Q\_S^\* S\_X \right\} \\\\ & - i \left\{ Q^\* \left[ -i\lambda\_1 x + a\_1(T) + \Lambda a\_2(T) \right] \right\}. \end{split}$$

In accordance with conditions (11), the imaginary functions *a*0(*T*), *a*1(*T*), *a*2(*T*) can be defined in the following way: *a*0(*T*) = *iγ*0(*T*), *a*1(*T*) = *iV*(*T*), *a*2(*T*) = −*iD*2(*T*), *R*2(*T*) = *F*2*γD*2(*T*),where *D*2(*T*), *V*(*T*), *γ*0(*T*) are arbitrary real functions. The coefficients *D*2(*T*) and *R*2(*T*) are represented by positively defined functions (for *σ* = −1, *γ* is assumed as a semi-entire number).

Then, Eqs. (9,10) can be transformed into

$$i\mathcal{Q}\_T = -\frac{1}{2}D\_2\mathcal{Q}\_{\text{SS}}\mathcal{S}\_x^2 - \sigma \mathcal{R}\_2 \left|\mathcal{Q}\right|^2 \mathcal{Q} - i\tilde{V}\mathcal{Q}\_{\text{S}} + i\Gamma \mathcal{Q} + \mathcal{U}\mathcal{Q}\_{\text{V}} \tag{12}$$

where

$$\tilde{V}(S,T) = \frac{1}{2}D\_2 S\_x^2 \rho\_S + V S\_x + S\_l - \lambda\_1 S\_r$$

$$M(S,T) = \frac{1}{8}D\_2 S\_x^2 \rho\_S^2 - 2\gamma\_0 + \frac{1}{2} \left(\varphi\_T + \varphi\_S S\_l + V S\_X \varphi\_S\right) + 2\lambda\_0 S / S\_x - \frac{1}{2} \lambda\_1 \varphi\_S S\_r \tag{13}$$

$$\Gamma = \left( -\gamma \frac{F\_T}{F} - \frac{1}{4} D\_2 S\_x^2 \varrho\_{\rm SS} + \lambda\_1 \right) = \left( \frac{1}{2} \frac{\mathcal{W}(R\_2, D\_2)}{R\_2 D\_2} - \frac{1}{4} D\_2 S\_x^2 \varrho\_{\rm SS} + \lambda\_1 \right). \tag{14}$$

Eq.(12) can be written down in the independent variables (*x*, *t*)

$$i\mathbf{Q}\_t + \frac{1}{2}D\_2(t)\mathbf{Q}\_{\mathbf{x}\mathbf{x}} + \sigma \mathbf{R}\_2(t) \left|\mathbf{Q}\right|^2 \mathbf{Q} - \mathcal{U}(\mathbf{x}, t)\mathbf{Q} + i\widetilde{\mathcal{V}}^\prime \mathbf{Q}\_{\mathbf{x}} = i\Gamma(t)\mathbf{Q}.\tag{15}$$

Let us transform Eq.(15) into the more convenient form

$$\mathrm{i}Q\_t + \frac{1}{2}D\_2Q\_{\mathrm{xx}} + \sigma R\_2 \left| Q \right|^2 Q - \mathrm{i}IQ = \mathrm{i}\Gamma Q \tag{16}$$

using the following condition

$$
\widetilde{V}' = \frac{1}{2} D\_2 \mathbf{S}\_x \boldsymbol{\varrho}\_S + V - \lambda\_1 \mathbf{S} / \mathbf{S}\_x = \mathbf{0}.\tag{17}
$$

If we apply the commonly accepted in the IST method (Ablowitz et al., 1973) reduction: *V* = −*ia*<sup>1</sup> = 0 , we find a parameter *λ*<sup>1</sup> from (17)

$$
\lambda\_1 = \frac{1}{2} D\_2 S\_\chi^2 \varphi\_S / S\_\prime \tag{18}
$$

and the corresponding potential *U*(*S*, *T*) from Eq.(13):

$$\mathcal{U}(S,T) = -2\gamma\_0 + 2\lambda\_0 S/S\_\mathcal{X} + \frac{1}{2} \left(\varphi\_T + \varphi\_S S\_l\right) - \frac{1}{8} D\_2 S\_\mathcal{X}^2 \rho\_\mathcal{S}^2. \tag{19}$$

According to Eq.(14), the gain or absorption coefficient now is represented by

$$
\Gamma = \frac{1}{2} \frac{W(R\_2, D\_2)}{R\_2 D\_2} - \frac{1}{4} D\_2 S\_x^2 \rho\_{SS} + \frac{1}{2} D\_2 S\_x^2 \rho\_S / S. \tag{20}
$$

Let us consider some special choices of variables to specify the solutions of (16). First of all, we assume that variables are factorized in the phase profile *ϕ*(*S*, *T*) as *ϕ* = *C*(*T*)*Sα*. The first term in the real potential (19) represents some additional time-dependent phase *e*2*γ*0(*t*)*<sup>t</sup>* of the solution *Q*(*x*, *t*) for the equation (16) and, without loss of the generality, we use *γ*<sup>0</sup> = 0. The second term in (19) depends linearly on *S*. The NLSE with the linear spatial potential and constant *λ*0, describing the case of Alfen waves propagation in plasmas, has been studied previously in Ref. (Chen, 1976). We will study the more general case of chirped solitons in the Section 4 of this Chapter. Now, taking into account three last terms in (19), we obtain

$$\mathcal{U}(\mathbf{S},T) = 2\lambda\_0 \mathbf{S} / \mathbf{S}\_{\mathbf{x}} + \frac{1}{2} \mathbf{C}\_T \mathbf{S}^{\mathbf{a}} + 1/2 a \mathbf{C} \mathbf{S}^{\mathbf{a}-1} \mathbf{S}\_l - \frac{1}{8} D\_2 \mathbf{C}^2 S\_{\mathbf{x}}^2 a^2 \mathbf{S}^{\mathbf{a}-2}. \tag{21}$$

The gain or absorption coefficient (20) becomes

$$\Gamma(T) = \frac{1}{2} \frac{W(R\_2, D\_2)}{R\_2 D\_2} + \frac{\kappa}{4} (3 - \kappa) D\_2 S\_x^2 C S^{\alpha - 2} \tag{22}$$

and Eq.(18) takes a form

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

**nonautonomous nonlinear and dispersive systems: nonautonomous solitons**

*<sup>a</sup>*2(*T*)*QϕSSx* <sup>−</sup> <sup>1</sup>

*<sup>a</sup>*2(*T*)*Q*∗*ϕSSx* <sup>−</sup> <sup>1</sup>

*<sup>x</sup>ϕ<sup>S</sup>* + *VSx* + *St* − *λ*1*S*,

*W*(*R*2, *D*2) *R*2*D*<sup>2</sup>

<sup>2</sup> *<sup>Q</sup>* <sup>−</sup> *<sup>U</sup>*(*x*, *<sup>t</sup>*)*<sup>Q</sup>* <sup>+</sup> *iV*

<sup>2</sup> (*ϕ<sup>T</sup>* <sup>+</sup> *<sup>ϕ</sup>SSt* <sup>+</sup> *VSxϕS*) <sup>+</sup> <sup>2</sup>*λ*0*S*/*Sx* <sup>−</sup> <sup>1</sup>

− 1 4 *D*2*S*<sup>2</sup>

*D*2*Sxϕ<sup>S</sup>* + *V* − *λ*1*S*/*Sx* = 0. (17)

2

2

<sup>2</sup> <sup>−</sup> *<sup>i</sup>λ*1*S*/*Sx*<sup>Λ</sup> <sup>+</sup> *<sup>a</sup>*1(*T*)<sup>Λ</sup> <sup>+</sup> *<sup>a</sup>*2(*T*)Λ2,

 +

*<sup>S</sup>Sx* 

<sup>2</sup> *<sup>Q</sup>* <sup>−</sup> *iVQ <sup>S</sup>* <sup>+</sup> *<sup>i</sup>*Γ*<sup>Q</sup>* <sup>+</sup> *UQ*, (12)

2

�*Qx* = *i*Γ(*t*)*Q*. (15)

*<sup>x</sup>ϕSS* + *λ*<sup>1</sup>

<sup>2</sup> *<sup>Q</sup>* <sup>−</sup> *UQ* <sup>=</sup> *<sup>i</sup>*Γ*<sup>Q</sup>* (16)

*<sup>x</sup>ϕS*/*S*, (18)

*λ*1*ϕSS*, (13)

. (14)

*a*2(*T*)*QSSx*

*a*2(*T*)*Q*∗

**3. Generalized nonlinear Schrödinger equation and solitary waves in**

2

*<sup>A</sup>* <sup>=</sup> <sup>−</sup>*iλ*0*S*/*Sx* <sup>+</sup> *<sup>a</sup>*0(*T*) <sup>−</sup> <sup>1</sup>

Then, Eqs. (9,10) can be transformed into

8 *D*2*S*<sup>2</sup> *xϕ*<sup>2</sup>

*iQt* + 1 2

using the following condition

*iQT* <sup>=</sup> <sup>−</sup><sup>1</sup>

2

*D*2*QSSS*<sup>2</sup>

*<sup>V</sup>*(*S*, *<sup>T</sup>*) = <sup>1</sup>

*<sup>S</sup>* − 2*γ*<sup>0</sup> +

Eq.(12) can be written down in the independent variables (*x*, *t*)

Let us transform Eq.(15) into the more convenient form

*iQt* + 1 2

> *V* � <sup>=</sup> <sup>1</sup> 2

−*ia*<sup>1</sup> = 0 , we find a parameter *λ*<sup>1</sup> from (17)

*D*2(*t*)*Qxx* + *σR*2(*t*)|*Q*|

2 *D*2*S*<sup>2</sup>

1

*<sup>x</sup>ϕSS* + *λ*<sup>1</sup>

semi-entire number).

*<sup>U</sup>*(*S*, *<sup>T</sup>*) = <sup>1</sup>

Γ = −*γ FT <sup>F</sup>* <sup>−</sup> <sup>1</sup> 4 *D*2*S*<sup>2</sup>

where

*<sup>B</sup>* <sup>=</sup> <sup>√</sup>*σF<sup>γ</sup>* exp (*iϕ*/2)

*<sup>C</sup>* <sup>=</sup> <sup>√</sup>*σF<sup>γ</sup>* exp (−*iϕ*/2)

Let us study a special case of the reduction procedure for Eqs. (9,10) when *a*<sup>3</sup> = 0

 − *i* 4

*i* {*Q* [−*iλ*1*S*/*Sx* + *a*1(*T*) + Λ*a*2(*T*)]} ,

−*i* {*Q*<sup>∗</sup> [−*iλ*1*x* + *a*1(*T*) + Λ*a*2(*T*)]} .

*<sup>a</sup>*2(*T*)*σF*2*<sup>γ</sup>* <sup>|</sup>*Q*<sup>|</sup>

 *i* 4

*<sup>x</sup>* − *σR*<sup>2</sup> |*Q*|

 = 1 2

*D*2*Qxx* + *σR*<sup>2</sup> |*Q*|

*<sup>λ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2 *D*2*S*<sup>2</sup>

If we apply the commonly accepted in the IST method (Ablowitz et al., 1973) reduction: *V* =

In accordance with conditions (11), the imaginary functions *a*0(*T*), *a*1(*T*), *a*2(*T*) can be defined in the following way: *a*0(*T*) = *iγ*0(*T*), *a*1(*T*) = *iV*(*T*), *a*2(*T*) = −*iD*2(*T*), *R*2(*T*) = *F*2*γD*2(*T*),where *D*2(*T*), *V*(*T*), *γ*0(*T*) are arbitrary real functions. The coefficients *D*2(*T*) and *R*2(*T*) are represented by positively defined functions (for *σ* = −1, *γ* is assumed as a

$$
\lambda\_1 = \frac{1}{2} D\_2 S\_x^2 \mathbb{C} a S^{a-2}.\tag{23}
$$

If we assume that the functions Γ(*T*) and *λ*1(*T*) depend only on *T* and do not depend on *S*, we conclude that *α* = 0 or *α* = 2.

The study of the soliton solutions of the nonautonomous NLSE with varying coefficients without time and space phase modulation (chirp) and corresponding to the case of *α* = 0 has been carried out in Ref. (Serkin & Belyaeva, 2001a;b)*.* Let us find here the solutions of Eq.(16) with chirp in the case of *α* = 2, *ϕ*(*S*, *T*) = *C*(*T*)*S*2. In this case, Eq. (18) becomes *λ*<sup>1</sup> = *D*2*S*<sup>2</sup> *xC*. Now, the real spatial-temporal potential (21) takes the form

$$\left[\mathcal{U}\left[\mathcal{S}(\mathbf{x},t),T\right]\right] = 2\lambda\_0 \mathcal{S}/\mathcal{S}\_{\mathbf{x}} + \frac{1}{2}\left(\mathcal{C}\_T - D\_2 \mathcal{S}\_{\mathbf{x}}^2 \mathcal{C}^2\right)\mathcal{S}^2 + \mathcal{C}\mathcal{S}\mathcal{S}\_{\mathbf{f}}$$

Consider the simplest option to choose the variable *S*(*x*, *t*) when the variables (*x*, *t*) are factorized: *S*(*x*, *t*) = *P*(*t*)*x*. In this case, all main characteristic functions: the phase modulation

$$
\varphi(\mathbf{x},t) = \Theta(t)\mathbf{x}^2,\tag{24}
$$

the real potential

$$\mathcal{U}(\mathbf{x},t) = 2\lambda\_0 \mathbf{x} + \frac{1}{2} \left(\Theta\_l - D\_2 \Theta^2\right) \mathbf{x}^2 \equiv 2\lambda\_0(t)\mathbf{x} + \frac{1}{2}\Omega^2(t)\mathbf{x}^2,\tag{25}$$

the gain (or absorption) coefficient

$$\Gamma(t) = \frac{1}{2} \left( \frac{\mathcal{W}(R\_2, D\_2)}{R\_2 D\_2} + D\_2 \mathcal{P}^2 \mathbb{C} \right) = \frac{1}{2} \left( \frac{\mathcal{W}(R\_2, D\_2)}{R\_2 D\_2} + D\_2 \Theta \right) \tag{26}$$

After the substitutions

*<sup>Q</sup>*(*x*, *<sup>t</sup>*) = *<sup>q</sup>*(*x*, *<sup>t</sup>*) exp �� *<sup>t</sup>*

*i ∂q <sup>∂</sup><sup>t</sup>* <sup>+</sup> 1 2 *D*(*t*) *∂*2*q <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

to the following exact integrability conditions

The self-induced soliton phase shift is given by

*R*(0) are defined by the initial conditions.

dispersion *D*(*t*) and nonlinearity *R*(*t*) changes.

<sup>Ω</sup>2(*t*)*D*(*t*) = *<sup>W</sup>*(*R*, *<sup>D</sup>*)

<sup>=</sup> *<sup>d</sup>*

and the time-dependent spectral parameter is represented by

<sup>Λ</sup>(*t*) = *<sup>κ</sup>*(*t*) + *<sup>i</sup>η*(*t*) = *<sup>D</sup>*0*R*(*t*)

0

Γ(*τ*)*dτ* �

*RD*

*dt* ln *<sup>D</sup>*(*t*) *<sup>d</sup>*

Eq. (28) is transformed to the generalized NLSE without gain or loss term

�

*σR*(*t*)|*q*|

*d*

, *<sup>R</sup>*(*t*) = *<sup>R</sup>*2(*t*) exp �

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 59

Finally, the Lax equation (2) with matrices (3-6) provides the nonautonomous model (30) under condition that dispersion *D*(*t*), nonlinearity *R*(*t*), and the harmonic potential satisfy

*dt* ln *<sup>R</sup>*(*t*) <sup>−</sup> *<sup>d</sup>*

*dt* ln *<sup>R</sup>*(*t*) <sup>−</sup> *<sup>d</sup>*<sup>2</sup>

<sup>Θ</sup>(*t*) = <sup>−</sup>*<sup>W</sup>* [(*R*(*t*), *<sup>D</sup>*(*t*)]

⎡

where the main parameters: time invariant eigenvalue Λ(0) = *κ*<sup>0</sup> + *iη*0; *D*<sup>0</sup> = *D*(0); *R*<sup>0</sup> =

We call Eq. (31) as the law of the soliton adaptation to the external potentials. The basic property of classical solitons to interact elastically holds true, but the novel feature of the nonautonomous solitons arises. Namely, both amplitudes and speeds of the solitons, and consequently, their spectra, during the propagation and after the interaction are no longer the same as those prior to the interaction. All nonautonomous solitons generally move with varying amplitudes *η*(*t*) and speeds *κ*(*t*) adapted both to the external potentials and to the

Having obtained the eigenvalue equations for scattering potential, we can write down the general solutions for bright (*σ* = +1) and dark (*σ* = −1) nonautonomous solitons applying

which connects the (*n* −1) and *n* - soliton solutions by means of the so-called pseudo-potential **<sup>Γ</sup>**�*n*−1(*x*, *<sup>t</sup>*) = *<sup>ψ</sup>*1(*x*, *<sup>t</sup>*)/*ψ*2(*x*, *<sup>t</sup>*) for the (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)−soliton scattering functions *<sup>ψ</sup>*(*x*, *<sup>t</sup>*)=(*ψ*1*ψ*2)*T*.

� � � 2 × � *D*(*t*)

the auto-Bäcklund transformation (Chen, 1974) and the recurrent relation

1 + � � �� **<sup>Γ</sup>***n*−1(*x*, *<sup>t</sup>*)

*qn*(*x*, *<sup>t</sup>*) = <sup>−</sup>*qn*−1(*x*, *<sup>t</sup>*) <sup>−</sup> <sup>4</sup>*ηn***Γ**�*n*−1(*x*, *<sup>t</sup>*)

<sup>⎣</sup>Λ(0) + *<sup>R</sup>*<sup>0</sup>

*D*<sup>0</sup> � *t*

0

*R*0*D*(*t*)

<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*0(*t*)*<sup>x</sup>* <sup>−</sup> <sup>1</sup>

2 � *t* 0

2

*dt* �*W*(*R*, *<sup>D</sup>*) *RD* �

*dt*<sup>2</sup> ln *<sup>D</sup>*(*t*) <sup>−</sup> *<sup>R</sup>*(*t*) *<sup>d</sup>*<sup>2</sup>

Γ(*τ*)*dτ* �

Ω2(*t*)*x*<sup>2</sup>

�

*dt*<sup>2</sup>

*<sup>D</sup>*2(*t*)*R*(*t*) (32)

*λ*0(*τ*)*D*(*τ*) *<sup>R</sup>*(*τ*) *<sup>d</sup><sup>τ</sup>*

1 *R*(*t*)

⎤

*<sup>R</sup>*(*t*) exp[−*i*Θ*x*2/2], (34)

, *D*(*t*) = *D*2(*t*),

*q* = 0. (30)

. (31)

⎦ , (33)

and the spectral parameter *λ*<sup>1</sup>

$$
\lambda\_1(t) = D\_2 P^2 \mathbb{C} = D\_2(t) \Theta(t) \tag{27}
$$

are found to be dependent on the self-induced soliton phase shift Θ(*t*). Notice that the definition <sup>Ω</sup>2(*t*) <sup>≡</sup> <sup>Θ</sup>*<sup>t</sup>* <sup>−</sup> *<sup>D</sup>*2Θ<sup>2</sup> has been introduced in Eq.(25).

Now we can rewrite the generalized NLSE (16) with time-dependent nonlinearity, dispersion and gain or absorption in the form of the nonautonomous NLSE with linear and parabolic potentials

$$\mathrm{i}Q\_t + \frac{1}{2}D\_2(t)Q\_{\mathrm{xx}} + \sigma \mathsf{R}\_2(t) \left| Q \right|^2 Q - 2\lambda\_0(t)\mathbf{x} - \frac{1}{2}\,\Omega^2(t)\mathbf{x}^2 Q = \mathrm{i}\Gamma Q. \tag{28}$$
