**2. Lax operator method and exact integrability of nonautonomous nonlinear and dispersive models with external potentials**

The classification of dynamic systems into autonomous and nonautonomous is commonly used in science to characterize different physical situations in which, respectively, external time-dependent driving force is being present or absent. The mathematical treatment of nonautonomous system of equations is much more complicated then of traditional autonomous ones. As a typical illustration we may mention both a simple pendulum whose length changes with time and parametrically driven nonlinear Duffing oscillator (Nayfeh & Balachandran, 2004).

In the framework of the IST method, the nonlinear integrable equation arises as the compatibility condition of the system of the linear matrix differential equations

$$
\psi\_{\mathbf{x}} = \widehat{\mathcal{F}}\psi(\mathbf{x}, t), \qquad \psi\_{\mathbf{t}} = \widehat{\mathcal{G}}\psi(\mathbf{x}, t). \tag{1}
$$

*<sup>C</sup>* <sup>=</sup> <sup>√</sup>*σF<sup>γ</sup>* exp[−*iϕS*/2]{− *<sup>i</sup>*

*a*2*RϕSSx* +

1 2

*a*3*RϕSSx* +

*a*3*QSSSS*<sup>3</sup>

<sup>−</sup>*St* <sup>+</sup> *<sup>λ</sup>*1*<sup>S</sup>* <sup>+</sup> *ia*1*Sx* <sup>−</sup> *<sup>i</sup>*

*iλ*<sup>1</sup> − *iγ*

*a*3*RSSSS*<sup>3</sup>

<sup>−</sup>*St* <sup>+</sup> *<sup>λ</sup>*1*<sup>S</sup>* <sup>+</sup> *ia*1*Sx* <sup>−</sup> *<sup>i</sup>*

*iλ*<sup>1</sup> − *iγ*

<sup>−</sup>2*λ*0*S*/*Sx* <sup>−</sup> <sup>2</sup>*ia*<sup>0</sup> <sup>−</sup> <sup>1</sup>

2*λ*0*S*/*Sx* + 2*ia*<sup>0</sup> +

+*Q i* 8 *a*2*ϕ*<sup>2</sup> *SS*2 *<sup>x</sup>* <sup>−</sup> *<sup>i</sup>*

−3 2

+*R* 

> +*R* − *i* 8 *a*2*ϕ*<sup>2</sup> *SS*2 *<sup>x</sup>* + *i* <sup>32</sup> *<sup>a</sup>*3*ϕ*<sup>3</sup> *SS*3 *<sup>x</sup>* <sup>−</sup> *<sup>i</sup>* 8

and (10) take the same form if the following conditions

*a*<sup>0</sup> = −*a*<sup>∗</sup>

*λ*<sup>0</sup> = *λ*∗

introduced within corresponding integrations.

*iRT* <sup>=</sup> <sup>1</sup> 4

− *i* 4

+Λ − *i* 4

+*iQS* 

> +*Q*

+*iRS* 

+*R* 

are fulfilled.

*iQT* <sup>=</sup> <sup>1</sup> 4

> −3 2

+*Q* 

and two general equations

4 *a*3*S*<sup>2</sup> *x* 

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

*a*3*RSSx* + *ia*2*R*

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 55

<sup>8</sup> *<sup>a</sup>*3*QSSϕSS*<sup>3</sup>

2 *a*2*ϕSS*<sup>2</sup> *<sup>x</sup>* + 3 8

2

*a*2*ϕSSS*<sup>2</sup>

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

<sup>32</sup> *<sup>a</sup>*3*ϕ*<sup>3</sup> *SS*3 *<sup>x</sup>* + *i* 8

<sup>8</sup> *<sup>a</sup>*3*RSSϕSS*<sup>3</sup>

2 *a*2*ϕSS*<sup>2</sup>

*i* 2

*a*2*ϕSSS*<sup>2</sup>

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

where the arbitrary time-dependent functions *a*<sup>0</sup> (*T*), *a*<sup>1</sup> (*T*), *a*<sup>2</sup> (*T*), *a*<sup>3</sup> (*T*) have been

By using the following reduction procedure *R* = −*Q*∗, it is easy to find that two equations (9)

<sup>1</sup>, *a*<sup>2</sup> = −*a*<sup>∗</sup>

<sup>1</sup>, *F* = *F*<sup>∗</sup>

1 2

*<sup>x</sup>* + 3*i*

*<sup>a</sup>*3*σF*2*<sup>γ</sup>QRQSSx* <sup>−</sup> *<sup>i</sup>*

*FT <sup>F</sup>* <sup>+</sup> 1 2

1

*<sup>x</sup>* <sup>−</sup> <sup>3</sup>*<sup>i</sup>*

*a*3*σF*2*γR*<sup>2</sup>*QSSx* +

*FT <sup>F</sup>* <sup>+</sup> 1 2

0, *a*<sup>1</sup> = −*a*<sup>∗</sup>

0, *λ*<sup>1</sup> = *λ*<sup>∗</sup>

*RSS* <sup>−</sup> *<sup>i</sup>* 2

−*iλ*1*S*/*Sx* +

*<sup>x</sup>* <sup>−</sup> <sup>3</sup>*<sup>i</sup>*

*a*2*QSSS*<sup>2</sup>

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

*<sup>x</sup>* + 3*i*

*a*2*RSSS*<sup>2</sup>

*<sup>x</sup>* <sup>−</sup> <sup>3</sup> 8

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

2

*<sup>R</sup>ϕSS* <sup>−</sup> <sup>1</sup> 4 *Rϕ*<sup>2</sup>

> 1 2

<sup>+</sup> *ia*3Λ2*R*},

*<sup>x</sup>* + *ia*2*σF*2*γQ*2*<sup>R</sup>*

*<sup>x</sup>* + 3*i* <sup>16</sup> *<sup>a</sup>*3*ϕ*<sup>2</sup> *SS*3 *x* 

> *x*

2 *a*1*ϕSSx*

*a*3*ϕSSS*<sup>3</sup>

<sup>16</sup> *<sup>a</sup>*3*ϕSϕSSS*<sup>3</sup>

*a*3*ϕSSSS*<sup>3</sup> *x* 

*<sup>λ</sup>*1*Sϕ<sup>S</sup>* <sup>−</sup> *<sup>i</sup>*

*<sup>x</sup>* <sup>−</sup> *ia*2*σF*2*γR*2*<sup>Q</sup>*

*<sup>x</sup>* + 3*i* <sup>16</sup> *<sup>a</sup>*3*ϕ*<sup>2</sup> *SS*3 *x* 

> *x*

> > *i* 2 *a*1*ϕSSx*

*a*3*ϕSSS*<sup>3</sup>

1 2

<sup>16</sup> *<sup>a</sup>*3*ϕSϕSSS*<sup>3</sup>

*λ*1*Sϕ<sup>S</sup>* +

*a*3*ϕSSSS*<sup>3</sup> *x* ,

2, *a*<sup>3</sup> = −*a*<sup>∗</sup>

*<sup>S</sup>* − *iRSϕ<sup>S</sup>*

<sup>4</sup> *<sup>a</sup>*3*σF*2*γQ*2*RϕSSx* (9)

<sup>4</sup> *<sup>a</sup>*3*σF*2*γR*2*QϕSSx* (10)

3, (11)

*a*3*σF*2*γQR* + *a*<sup>1</sup>

Here *<sup>ψ</sup>*(*x*, *<sup>t</sup>*) = {*ψ*1, *<sup>ψ</sup>*2}*<sup>T</sup>* is a 2-component complex function, <sup>F</sup> and <sup>G</sup> are complex-valued (2 × 2) matrices. Let us consider the general case of the IST method with a time-dependent spectral parameter Λ(*T*) and the matrices F and G

$$
\hat{\mathcal{F}}(\Lambda; S, T) = \hat{\mathcal{F}}\left\{\Lambda(T), \boldsymbol{q}\left[S(\mathbf{x}, t), T\right]; \frac{\partial \boldsymbol{q}}{\partial S}\left(\frac{\partial S}{\partial \mathbf{x}}\right); \frac{\partial^2 \boldsymbol{q}}{\partial S^2}\left(\frac{\partial S}{\partial \mathbf{x}}\right)^2; \dots; \frac{\partial^n \boldsymbol{q}}{\partial S^n}\left(\frac{\partial S}{\partial \mathbf{x}}\right)^n\right\};
$$

$$
\hat{\mathcal{G}}(\Lambda; S, T) = \hat{\mathcal{G}}\left\{\Lambda(T), \boldsymbol{q}\left[S(\mathbf{x}, t), T\right]; \frac{\partial \boldsymbol{q}}{\partial S}\left(\frac{\partial S}{\partial \mathbf{x}}\right); \frac{\partial^2 \boldsymbol{q}}{\partial S^2}\left(\frac{\partial S}{\partial \mathbf{x}}\right)^2; \dots; \frac{\partial^n \boldsymbol{q}}{\partial S^n}\left(\frac{\partial S}{\partial \mathbf{x}}\right)^n\right\},
$$

dependent on the generalized coordinates *S* = *S*(*x*, *t*) and *T*(*t*) = *t*, where the function *q* [*S*(*x*, *t*), *T*] and its derivatives denote the scattering potentials *Q*(*S*, *T*) and *R*(*S*, *T*) and their derivatives, correspondingly. The condition for the compatibility of the pair of linear differential equations (1) takes a form

$$\frac{\partial \widehat{\mathcal{F}}}{\partial T} + \frac{\partial \widehat{\mathcal{F}}}{\partial \mathcal{S}} \mathbf{S}\_{l} - \frac{\partial \widehat{\mathcal{G}}}{\partial \mathcal{S}} \mathbf{S}\_{x} + \left[\widehat{\mathcal{F}}, \widehat{\mathcal{G}}\right] = \mathbf{0},\tag{2}$$

where

$$
\hat{\mathcal{F}} = -i\Lambda(T)\hat{\sigma}\_3 + \hat{\mathcal{U}}\hat{\Phi}\_{\prime} \tag{3}
$$

$$
\hat{\mathcal{G}} = \begin{pmatrix} A & B \\ \mathcal{C} & -A \end{pmatrix} \tag{4}
$$

*<sup>σ</sup>*<sup>3</sup> is the Pauli spin matrix and matrices <sup>U</sup> and *<sup>φ</sup>* are given by

$$
\hat{\mathcal{U}} = \sqrt{\sigma} F^{\gamma} \begin{pmatrix} T \\ \end{pmatrix} \begin{pmatrix} 0 & Q(\mathcal{S}, T) \\ \mathcal{R}(\mathcal{S}, T) & 0 \\ \end{pmatrix} . \tag{5}
$$

$$
\hat{\phi} = \begin{pmatrix}
\exp[-i\varphi/2] & 0 \\
0 & \exp[i\varphi/2]
\end{pmatrix}.
\tag{6}
$$

Here *F*(*T*) and *ϕ*(*S*, *T*) are real unknown functions, *γ* is an arbitrary constant, and *σ* = ±1. The desired elements of G matrix (known in the modern literature as the AKNS elements) can be constructed in the form <sup>G</sup> <sup>=</sup> <sup>∑</sup>*k*=<sup>3</sup> *<sup>k</sup>*=<sup>0</sup> *Gk*Λ*k*,with time varying spectral parameter given by

$$
\Lambda\_T = \lambda\_0 \left( T \right) + \lambda\_1 \left( T \right) \Lambda \left( T \right), \tag{7}
$$

where time-dependent functions *λ*<sup>0</sup> (*T*) and *λ*<sup>1</sup> (*T*) are the expansion coefficients of Λ*<sup>T</sup>* in powers of the spectral parameter Λ (*T*).

Solving the system (2-6), we find both the matrix elements *A*, *B*, *C*

$$A = -i\lambda\_0 S/S\_X + a\_0 - \frac{1}{4}a\_3 \sigma F^{2\gamma} (QR\varphi\_S S\_X + iQR\_S S\_X - iRQ\_S S\_X) \tag{8}$$

$$+ \frac{1}{2}a\_2 \sigma F^{2\gamma} QR + \Lambda \left( -i\lambda\_1 S/S\_X + \frac{1}{2}a\_3 \sigma F^{2\gamma} QR + a\_1 \right) + a\_2 \Lambda^2 + a\_3 \Lambda^3,$$

$$B = \sqrt{\sigma} F^{\gamma} \exp[i g S/2] \left\{ -\frac{i}{4} a\_3 S\_X^2 \left( Q\_{SS} + \frac{i}{2} Q g\_{SS} - \frac{1}{4} Q g\_S^2 + iQ\_S g\_S \right) \right.$$

$$- \frac{i}{4} a\_2 Q \varphi\_S S\_X - \frac{1}{2} a\_2 Q\_S S\_X + iQ \left( -i\lambda\_1 S/S\_X + \frac{1}{2} a\_3 \sigma F^{2\gamma} QR + a\_1 \right)$$

$$+ \Lambda \left( -\frac{i}{4} a\_3 Q \varphi\_S S\_X - \frac{1}{2} a\_3 Q\_S S\_X + ia\_2 Q \right) + i a\_3 \Lambda^2 Q \},$$

$$\begin{split} \mathcal{C} &= \sqrt{\sigma} F^{\gamma} \exp \left[ -i\varrho S / 2 \right] \{ -\frac{i}{4} a\_3 S\_x^2 \left( R\_{SS} - \frac{i}{2} R \varrho\_{SS} - \frac{1}{4} R \varrho\_S^2 - i R\_S \varrho\_S \right) \} \\ &- \frac{i}{4} a\_2 R \varrho\_S S\_X + \frac{1}{2} a\_2 R\_S S\_X + i R \left( -i \lambda\_1 S / S\_x + \frac{1}{2} a\_3 \sigma F^{2\gamma} QR + a\_1 \right) \\ &+ \Lambda \left( -\frac{i}{4} a\_3 R \varrho\_S S\_X + \frac{1}{2} a\_3 R\_S S\_X + i a\_2 R \right) + i a\_3 \Lambda^2 R \} , \end{split}$$

and two general equations

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

Here *<sup>ψ</sup>*(*x*, *<sup>t</sup>*) = {*ψ*1, *<sup>ψ</sup>*2}*<sup>T</sup>* is a 2-component complex function, <sup>F</sup> and <sup>G</sup> are complex-valued (2 × 2) matrices. Let us consider the general case of the IST method with a time-dependent

> *∂q ∂S*

*∂q ∂S*  *∂S ∂x* ; *∂*2*q ∂S*<sup>2</sup>

 *∂S ∂x* ; *∂*2*q ∂S*<sup>2</sup>

dependent on the generalized coordinates *S* = *S*(*x*, *t*) and *T*(*t*) = *t*, where the function *q* [*S*(*x*, *t*), *T*] and its derivatives denote the scattering potentials *Q*(*S*, *T*) and *R*(*S*, *T*) and their derivatives, correspondingly. The condition for the compatibility of the pair of linear

*<sup>∂</sup><sup>S</sup> Sx* <sup>+</sup>

 *A B C* −*A*

exp[−*iϕ*/2] <sup>0</sup>

Here *F*(*T*) and *ϕ*(*S*, *T*) are real unknown functions, *γ* is an arbitrary constant, and *σ* = ±1. The desired elements of G matrix (known in the modern literature as the AKNS elements) can

where time-dependent functions *λ*<sup>0</sup> (*T*) and *λ*<sup>1</sup> (*T*) are the expansion coefficients of Λ*<sup>T</sup>* in

1 2

*a*3*QSSx* + *ia*2*Q*

*i* 2

−*iλ*1*S*/*Sx* +

 F, G 

 0 *Q*(*S*, *T*) *R*(*S*, *T*) 0

0 exp[*iϕ*/2]

 *∂S ∂x* <sup>2</sup> ; ...; *∂nq ∂S<sup>n</sup>*

 *∂S ∂x* <sup>2</sup> ; ...; *∂nq ∂S<sup>n</sup>*

<sup>F</sup> <sup>=</sup> <sup>−</sup>*i*Λ(*T*)*<sup>σ</sup>*<sup>3</sup> <sup>+</sup> <sup>U</sup>*<sup>φ</sup>*, (3)

*<sup>k</sup>*=<sup>0</sup> *Gk*Λ*k*,with time varying spectral parameter given by Λ*<sup>T</sup>* = *λ*<sup>0</sup> (*T*) + *λ*<sup>1</sup> (*T*) Λ (*T*), (7)

*<sup>a</sup>*3*σF*2*γ*(*QRϕSSx* <sup>+</sup> *iQRSSx* <sup>−</sup> *iRQSSx*) (8)

+ *a*2Λ<sup>2</sup> + *a*3Λ3,

*<sup>S</sup>* + *iQSϕ<sup>S</sup>*

*a*3*σF*2*γQR* + *a*<sup>1</sup>

*a*3*σF*2*γQR* + *a*<sup>1</sup>

*<sup>Q</sup>ϕSS* <sup>−</sup> <sup>1</sup> 4 *Qϕ*<sup>2</sup>

> 1 2

<sup>+</sup> *ia*3Λ2*Q*},

 *∂S ∂x*

 *∂S ∂x*

= 0, (2)

, (5)

. (6)

, (4)

*<sup>n</sup>* 

*<sup>n</sup>* ,

spectral parameter Λ(*T*) and the matrices F and G

Λ(*T*), *q* [*S*(*x*, *t*), *T*] ;

Λ(*T*), *q* [*S*(*x*, *t*), *T*] ;

*∂*F *<sup>∂</sup><sup>T</sup>* <sup>+</sup>

*<sup>σ</sup>*<sup>3</sup> is the Pauli spin matrix and matrices <sup>U</sup> and *<sup>φ</sup>* are given by

<sup>U</sup> <sup>=</sup> <sup>√</sup>*σF<sup>γ</sup>* (*T*)

*φ* =

<sup>G</sup> <sup>=</sup> <sup>∑</sup>*k*=<sup>3</sup>

Solving the system (2-6), we find both the matrix elements *A*, *B*, *C*

4

2

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

−*iλ*1*S*/*Sx* +

4 *a*3*S*<sup>2</sup> *x QSS* +

*a*2*QSSx* + *iQ*

2

*∂*F *<sup>∂</sup><sup>S</sup> St* <sup>−</sup> *<sup>∂</sup>*<sup>G</sup>

G =

F(Λ; *S*, *T*)=F

G(Λ; *S*, *T*)=G

be constructed in the form

+ 1 2

> − *i* 4

+Λ − *i* 4

powers of the spectral parameter Λ (*T*).

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

*a*2*σF*2*γQR* + Λ

*<sup>B</sup>* <sup>=</sup> <sup>√</sup>*σF<sup>γ</sup>* exp[*iϕS*/2]{− *<sup>i</sup>*

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

where

differential equations (1) takes a form

*iQT* <sup>=</sup> <sup>1</sup> 4 *a*3*QSSSS*<sup>3</sup> *<sup>x</sup>* + 3*i* <sup>8</sup> *<sup>a</sup>*3*QSSϕSS*<sup>3</sup> *<sup>x</sup>* <sup>−</sup> <sup>3</sup>*<sup>i</sup>* <sup>4</sup> *<sup>a</sup>*3*σF*2*γQ*2*RϕSSx* (9) −3 2 *<sup>a</sup>*3*σF*2*<sup>γ</sup>QRQSSx* <sup>−</sup> *<sup>i</sup>* 2 *a*2*QSSS*<sup>2</sup> *<sup>x</sup>* + *ia*2*σF*2*γQ*2*<sup>R</sup>* +*iQS* <sup>−</sup>*St* <sup>+</sup> *<sup>λ</sup>*1*<sup>S</sup>* <sup>+</sup> *ia*1*Sx* <sup>−</sup> *<sup>i</sup>* 2 *a*2*ϕSS*<sup>2</sup> *<sup>x</sup>* + 3 8 *a*3*ϕSSS*<sup>3</sup> *<sup>x</sup>* + 3*i* <sup>16</sup> *<sup>a</sup>*3*ϕ*<sup>2</sup> *SS*3 *x* +*Q iλ*<sup>1</sup> − *iγ FT <sup>F</sup>* <sup>+</sup> 1 2 *a*2*ϕSSS*<sup>2</sup> *<sup>x</sup>* <sup>−</sup> <sup>3</sup> <sup>16</sup> *<sup>a</sup>*3*ϕSϕSSS*<sup>3</sup> *x* +*Q* 2*λ*0*S*/*Sx* + 2*ia*<sup>0</sup> + 1 <sup>2</sup> (*ϕ<sup>T</sup>* <sup>+</sup> *<sup>ϕ</sup>SSt*) <sup>−</sup> <sup>1</sup> 2 *<sup>λ</sup>*1*Sϕ<sup>S</sup>* <sup>−</sup> *<sup>i</sup>* 2 *a*1*ϕSSx* +*Q i* 8 *a*2*ϕ*<sup>2</sup> *SS*2 *<sup>x</sup>* <sup>−</sup> *<sup>i</sup>* <sup>32</sup> *<sup>a</sup>*3*ϕ*<sup>3</sup> *SS*3 *<sup>x</sup>* + *i* 8 *a*3*ϕSSSS*<sup>3</sup> *x iRT* <sup>=</sup> <sup>1</sup> 4 *a*3*RSSSS*<sup>3</sup> *<sup>x</sup>* <sup>−</sup> <sup>3</sup>*<sup>i</sup>* <sup>8</sup> *<sup>a</sup>*3*RSSϕSS*<sup>3</sup> *<sup>x</sup>* + 3*i* <sup>4</sup> *<sup>a</sup>*3*σF*2*γR*2*QϕSSx* (10) −3 2 *a*3*σF*2*γR*<sup>2</sup>*QSSx* + *i* 2 *a*2*RSSS*<sup>2</sup> *<sup>x</sup>* <sup>−</sup> *ia*2*σF*2*γR*2*<sup>Q</sup>* +*iRS* <sup>−</sup>*St* <sup>+</sup> *<sup>λ</sup>*1*<sup>S</sup>* <sup>+</sup> *ia*1*Sx* <sup>−</sup> *<sup>i</sup>* 2 *a*2*ϕSS*<sup>2</sup> *<sup>x</sup>* <sup>−</sup> <sup>3</sup> 8 *a*3*ϕSSS*<sup>3</sup> *<sup>x</sup>* + 3*i* <sup>16</sup> *<sup>a</sup>*3*ϕ*<sup>2</sup> *SS*3 *x* +*R iλ*<sup>1</sup> − *iγ FT <sup>F</sup>* <sup>+</sup> 1 2 *a*2*ϕSSS*<sup>2</sup> *<sup>x</sup>* <sup>−</sup> <sup>3</sup> <sup>16</sup> *<sup>a</sup>*3*ϕSϕSSS*<sup>3</sup> *x* +*R* <sup>−</sup>2*λ*0*S*/*Sx* <sup>−</sup> <sup>2</sup>*ia*<sup>0</sup> <sup>−</sup> <sup>1</sup> <sup>2</sup> (*ϕ<sup>T</sup>* <sup>+</sup> *<sup>ϕ</sup>SSt*) <sup>+</sup> 1 2 *λ*1*Sϕ<sup>S</sup>* + *i* 2 *a*1*ϕSSx* +*R* − *i* 8 *a*2*ϕ*<sup>2</sup> *SS*2 *<sup>x</sup>* + *i* <sup>32</sup> *<sup>a</sup>*3*ϕ*<sup>3</sup> *SS*3 *<sup>x</sup>* <sup>−</sup> *<sup>i</sup>* 8 *a*3*ϕSSSS*<sup>3</sup> *x* ,

where the arbitrary time-dependent functions *a*<sup>0</sup> (*T*), *a*<sup>1</sup> (*T*), *a*<sup>2</sup> (*T*), *a*<sup>3</sup> (*T*) have been introduced within corresponding integrations.

By using the following reduction procedure *R* = −*Q*∗, it is easy to find that two equations (9) and (10) take the same form if the following conditions

$$\begin{aligned} a\_0 &= -a\_0^\*, \ a\_1 = -a\_1^\*, \ a\_2 = -a\_2^\*, \ a\_3 = -a\_3^\*, \\ \lambda\_0 &= \lambda\_0^\*, \quad \lambda\_1 = \lambda\_1^\*, \quad F = F^\* \end{aligned} \tag{11}$$

are fulfilled.

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

<sup>Γ</sup> <sup>=</sup> <sup>1</sup> 2

*U*(*S*, *T*) = 2*λ*0*S*/*Sx* +

The gain or absorption coefficient (20) becomes

and Eq.(18) takes a form

*λ*<sup>1</sup> = *D*2*S*<sup>2</sup>

modulation

the real potential

we conclude that *α* = 0 or *α* = 2.

<sup>Γ</sup>(*T*) = <sup>1</sup> 2

*U* [*S*(*x*, *t*), *T*)] = 2*λ*0*S*/*Sx* +

1 2 

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

*U*(*x*, *t*) = 2*λ*0*x* +

the gain (or absorption) coefficient

<sup>Γ</sup>(*t*) = <sup>1</sup> 2

*U*(*S*, *T*) = −2*γ*<sup>0</sup> + 2*λ*0*S*/*Sx* +

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

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

− 1 4 *D*2*S*<sup>2</sup>

Section 4 of this Chapter. Now, taking into account three last terms in (19), we obtain

1 2

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

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

*xC*. Now, the real spatial-temporal potential (21) takes the form

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

*CTS<sup>α</sup>* <sup>+</sup> 1/2*αCSα*−1*St* <sup>−</sup> <sup>1</sup>

(<sup>3</sup> <sup>−</sup> *<sup>α</sup>*)*D*2*S*<sup>2</sup>

+ *α* 4

If we assume that the functions Γ(*T*) and *λ*1(*T*) depend only on *T* and do not depend on *S*,

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

> 1 2

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

 <sup>=</sup> <sup>1</sup> 2

<sup>Θ</sup>*<sup>t</sup>* <sup>−</sup> *<sup>D</sup>*2Θ<sup>2</sup>

+ *D*2*P*2*C*

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

*xC*<sup>2</sup> 

*<sup>x</sup>*<sup>2</sup> <sup>≡</sup> <sup>2</sup>*λ*0(*t*)*<sup>x</sup>* <sup>+</sup>

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

1

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 57

*<sup>x</sup>ϕSS* +

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

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

8

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

*xCαSα*−2. (23)

*S*<sup>2</sup> + *CSSt*

*ϕ*(*x*, *t*) = Θ(*t*)*x*2, (24)

1 2

+ *D*2Θ

Ω2(*t*)*x*2, (25)

(26)

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

*<sup>S</sup>*. (19)

*<sup>x</sup>ϕS*/*S*. (20)

*<sup>x</sup>α*2*S*2*α*−2. (21)

*xCS<sup>α</sup>*−<sup>2</sup> (22)
