**4. Hidden features of the soliton adaptation law to external potentials: the generalized Serkin-Hasegawa theorems**

It is now generally accepted that solitary waves in nonautonomous nonlinear and dispersive systems can propagate in the form of so-called nonautonomous solitons or solitonlike similaritons (see (Atre et al., 2006; Avelar et al., 2009; Beli´c et al., 2008; Chen et al., 2007; Hao, 2008; He et al., 2009; Hernandez et al., 2005; Hernandez-Tenorio et al., 2007; Liu et al., 2008; Porsezian et al., 2009; 2007; Serkin et al., 2007; Shin, 2008; Tenorio et al., 2005; Wang et al., 2008; Wu, Li & Zhang, 2008; Wu, Zhang, Li, Finot & Porsezian, 2008; Zhang et al., 2008; Zhao et al., 2009; 2008) and references therein). Nonautonomous solitons interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The existence of specific laws of soliton adaptation to external gain and loss potentials was predicted by Serkin and Hasegawa in 2000 (Serkin & Hasegawa, 2000a;b; 2002). The physical mechanism resulting in the soliton stabilization in nonautonomous and dispersive systems was revealed in this paper. From the physical point of view, the adaptation means that solitons remain self similar and do not emit dispersive waves both during their interactions with external potentials and with each other. The soliton adaptation laws are known today as the Serkin-Hasegawa theorems (SH theorems). Serkin and Hasegawa obtained their SH-theorems by using the symmetry reduction methods when the initial nonautonomous NLSE can be transformed by the canonical autonomous NLSE under specific conditions found in (Serkin & Hasegawa, 2000a;b). Later, SH-theorems have been confirmed by different methods, in particular, by the Painleve analysis and similarity transformations (Serkin & Hasegawa, 2000a;b; 2002; Serkin et al., 2004; 2007; 2001a;b).

Substituting the phase profile Θ(*t*) given by Eq. (26) into Eq. (25), it is straightforward to verify that the frequency of the harmonic potential Ω(*t*) is related with dispersion *D*2(*t*), nonlinearity *R*2(*t*) and gain or absorption coefficient Γ(*t*) by the following conditions

$$\begin{split} \Omega^2(t)D\_2(t) &= D\_2(t)\frac{d}{dt}\left(\frac{\Gamma(t)}{D\_2(t)}\right) - \Gamma^2(t) \\ &\quad - \frac{d}{dt}\left(\frac{W(\mathcal{R}\_{2\prime}D\_2)}{R\_2D\_2}\right) + \left(2\Gamma(t) + \frac{d}{dt}\ln\mathcal{R}\_2(t)\right)\frac{W(\mathcal{R}\_{2\prime}D\_2)}{R\_2D\_2} \\ &= D\_2(t)\frac{d}{dt}\left(\frac{\Gamma(t)}{D\_2(t)}\right) - \Gamma^2(t) + \left(2\Gamma(t) + \frac{d}{dt}\ln\mathcal{R}\_2(t)\right)\frac{d}{dt}\ln\frac{D\_2(t)}{R\_2(t)} - \frac{d^2}{dt^2}\ln\frac{D\_2(t)}{R\_2(t)}, \end{split} \tag{29}$$

where *W*(*R*2, *D*) = *R*2*D*� <sup>2</sup>*<sup>t</sup>* <sup>−</sup> *<sup>D</sup>*2*R*� <sup>2</sup>*<sup>t</sup>* is the Wronskian. After the substitutions

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

are found to be dependent on the self-induced soliton phase shift Θ(*t*). Notice that the

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

It is now generally accepted that solitary waves in nonautonomous nonlinear and dispersive systems can propagate in the form of so-called nonautonomous solitons or solitonlike similaritons (see (Atre et al., 2006; Avelar et al., 2009; Beli´c et al., 2008; Chen et al., 2007; Hao, 2008; He et al., 2009; Hernandez et al., 2005; Hernandez-Tenorio et al., 2007; Liu et al., 2008; Porsezian et al., 2009; 2007; Serkin et al., 2007; Shin, 2008; Tenorio et al., 2005; Wang et al., 2008; Wu, Li & Zhang, 2008; Wu, Zhang, Li, Finot & Porsezian, 2008; Zhang et al., 2008; Zhao et al., 2009; 2008) and references therein). Nonautonomous solitons interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The existence of specific laws of soliton adaptation to external gain and loss potentials was predicted by Serkin and Hasegawa in 2000 (Serkin & Hasegawa, 2000a;b; 2002). The physical mechanism resulting in the soliton stabilization in nonautonomous and dispersive systems was revealed in this paper. From the physical point of view, the adaptation means that solitons remain self similar and do not emit dispersive waves both during their interactions with external potentials and with each other. The soliton adaptation laws are known today as the Serkin-Hasegawa theorems (SH theorems). Serkin and Hasegawa obtained their SH-theorems by using the symmetry reduction methods when the initial nonautonomous NLSE can be transformed by the canonical autonomous NLSE under specific conditions found in (Serkin & Hasegawa, 2000a;b). Later, SH-theorems have been confirmed by different methods, in particular, by the Painleve analysis and similarity transformations (Serkin & Hasegawa, 2000a;b; 2002; Serkin

Substituting the phase profile Θ(*t*) given by Eq. (26) into Eq. (25), it is straightforward to verify that the frequency of the harmonic potential Ω(*t*) is related with dispersion *D*2(*t*),

<sup>−</sup> <sup>Γ</sup>2(*t*)

<sup>2</sup>Γ(*t*) + *<sup>d</sup>*

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

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

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

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

*d*

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

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

(29)

nonlinearity *R*2(*t*) and gain or absorption coefficient Γ(*t*) by the following conditions

 + 

<sup>2</sup>Γ(*t*) + *<sup>d</sup>*

<sup>2</sup>*<sup>t</sup>* is the Wronskian.

 Γ(*t*) *D*2(*t*)

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

**4. Hidden features of the soliton adaptation law to external potentials: the**

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

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).

*D*2(*t*)*Qxx* + *σR*2(*t*)|*Q*|

**generalized Serkin-Hasegawa theorems**

*λ*1(*t*) = *D*2*P*2*C* = *D*2(*t*)Θ(*t*) (27)

<sup>2</sup> <sup>Ω</sup>2(*t*)*x*2*<sup>Q</sup>* <sup>=</sup> *<sup>i</sup>*Γ*Q*. (28)

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

*iQt* + 1 2

et al., 2004; 2007; 2001a;b).

<sup>=</sup> *<sup>D</sup>*2(*t*) *<sup>d</sup>*

where *W*(*R*2, *D*) = *R*2*D*�

*dt*

<sup>Ω</sup>2(*t*)*D*2(*t*) = *<sup>D</sup>*2(*t*) *<sup>d</sup>*

 Γ(*t*) *D*2(*t*) *dt*

<sup>−</sup> <sup>Γ</sup>2(*t*) +

− *d dt*

<sup>2</sup>*<sup>t</sup>* <sup>−</sup> *<sup>D</sup>*2*R*�

potentials

$$Q(\mathbf{x},t) = q(\mathbf{x},t) \exp\left[\int\_0^t \Gamma(\tau)d\tau\right], \text{ } \mathcal{R}(t) = \mathcal{R}\_2(t) \exp\left[2\int\_0^t \Gamma(\tau)d\tau\right], \text{ } D(t) = D\_2(t),$$

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

$$i\frac{\partial q}{\partial t} + \frac{1}{2}D(t)\frac{\partial^2 q}{\partial \mathbf{x}^2} + \left[\sigma \mathbf{R}(t)|q|^2 - 2\lambda\_0(t)\mathbf{x} - \frac{1}{2}\boldsymbol{\Omega}^2(t)\mathbf{x}^2\right]q = 0. \tag{30}$$

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 to the following exact integrability conditions

$$\begin{split} \Omega^2(t)D(t) &= \frac{W(R,D)}{RD}\frac{d}{dt}\ln R(t) - \frac{d}{dt}\left(\frac{\mathcal{W}(R,D)}{RD}\right) \\ &= \frac{d}{dt}\ln D(t)\frac{d}{dt}\ln R(t) - \frac{d^2}{dt^2}\ln D(t) - R(t)\frac{d^2}{dt^2}\frac{1}{R(t)}.\end{split} \tag{31}$$

The self-induced soliton phase shift is given by

$$\Theta(t) = -\frac{W\left[ (R(t), D(t)) \right]}{D^2(t)R(t)} \tag{32}$$

and the time-dependent spectral parameter is represented by

$$
\Lambda(t) = \kappa(t) + i\eta(t) = \frac{D\_0 \mathcal{R}(t)}{R\_0 D(t)} \left[ \Lambda(0) + \frac{R\_0}{D\_0} \int\_0^t \frac{\lambda\_0(\tau) D(\tau)}{\mathcal{R}(\tau)} d\tau \right], \tag{33}
$$

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

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

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

$$q\_n(\mathbf{x}, t) = -q\_{n-1}(\mathbf{x}, t) - \frac{4\eta\_n \tilde{\Gamma}\_{n-1}(\mathbf{x}, t)}{1 + \left| \tilde{\Gamma}\_{n-1}(\mathbf{x}, t) \right|^2} \times \sqrt{\frac{D(t)}{R(t)}} \exp[-i\Theta \mathbf{x}^2/2],\tag{34}$$

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*.

Two-soliton *q*2(*x*, *t*) solution for *σ* = +1 follows from Eq. (34)

� *D*(*t*) *R*(*t*)

<sup>×</sup>[(*κ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*1)<sup>2</sup> <sup>+</sup> <sup>2</sup>*iη*2(*κ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*1)tanh *<sup>ξ</sup>*<sup>2</sup> <sup>+</sup> *<sup>η</sup>*<sup>2</sup>

<sup>×</sup>[(*κ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*1)<sup>2</sup> <sup>−</sup> <sup>2</sup>*iη*1(*κ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*1)tanh *<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> *<sup>η</sup>*<sup>2</sup>

(*κ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*1)<sup>2</sup> <sup>+</sup> (*η*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*1)

(*κ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*1)<sup>2</sup> <sup>+</sup> (*η*<sup>2</sup> <sup>+</sup> *<sup>η</sup>*1)

� *t*

0

� *t*

*D*(*τ*) � *κ*2 *<sup>i</sup>* (*τ*) <sup>−</sup> *<sup>η</sup>*<sup>2</sup>

*R*0 *D*<sup>0</sup> � *t*

of the nonautonomous solitons, where *κ*0*<sup>i</sup>* and *η*0*<sup>i</sup>* correspond to the initial velocity and

Eqs. (39-45) describe the dynamics of two bounded solitons at all times and all locations. Obviously, these soliton solutions reduce to classical soliton solutions in the limit of autonomous nonlinear and dispersive systems given by conditions: *R*(*t*) = *D*(*t*) = 1, and

0

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

0

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

> ⎡ ⎣*κ*0*<sup>i</sup>* +

�

�

*ξi*(*x*, *t*) = 2*ηi*(*t*)*x* + 4

*χi*(*x*, *t*) = 2*κi*(*t*)*x* + 2

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

*λ*0(*t*) = Ω(*t*) ≡ 0 for canonical NLSE without external potentials.

N (*x*, *t*) <sup>D</sup> (*x*, *<sup>t</sup>*) exp �

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 61

− *i* 2 Θ(*t*)*x*<sup>2</sup> �

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

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

2 �

2 � , (39)

<sup>2</sup> ] + *η*<sup>2</sup> cosh *ξ*<sup>1</sup> exp (−*iχ*2)

<sup>2</sup> ], (40)

− 4*η*1*η*<sup>2</sup> cos (*χ*<sup>2</sup> − *χ*1). (41)

*dτ* (43)

⎦ (45)

*D*(*τ*)*ηi*(*τ*)*κi*(*τ*)*dτ*, (42)

*η*0*i*, (44)

⎤

*<sup>i</sup>* (*τ*) �

*q*2(*x*, *t*) = 4

where the numerator N (*x*, *t*) is given by

N = cosh *ξ*<sup>2</sup> exp (−*iχ*1)

and the denominator D (*x*, *t*) is represented by

+ cosh(*ξ*<sup>1</sup> − *ξ*2)

D = cosh(*ξ*<sup>1</sup> + *ξ*2)

Arguments and phases in Eqs.(39-41)

are related with the amplitudes

amplitude of the *i* -th soliton (*i* = 1, 2).

and velocities

Bright *q*<sup>+</sup> <sup>1</sup> (*x*, *t*) and dark *q*<sup>−</sup> <sup>1</sup> (*x*, *t*) soliton solutions are represented by the following analytic expressions:

$$q\_1^+(\mathbf{x}, t \mid \sigma = +1) = 2\eta\_1(t)\sqrt{\frac{D(t)}{R(t)}}\text{sech}\left[\tilde{\varrho}\_1(\mathbf{x}, t)\right] \times \exp\left\{-i\left(\frac{\Theta(t)}{2}\mathbf{x}^2 + \chi\_1(\mathbf{x}, t)\right)\right\};\tag{35}$$

$$q\_1^-(\mathbf{x}, t \mid \sigma = -1) = 2\eta\_1(t)\sqrt{\frac{D(t)}{R(t)}} \left[\sqrt{(1 - a^2)} + ia \tanh \zeta(\mathbf{x}, t)\right] \tag{36}$$

$$\times \exp\left\{-i\left(\frac{\Theta(t)}{2}\mathbf{x}^2 + \phi(\mathbf{x}, t)\right)\right\},$$

$$
\zeta(\mathbf{x},t) = 2a\eta\_1(t)\mathbf{x} + 4a \int\_0^t D(\tau)\eta\_1(\tau)\kappa\_1(\tau)d\tau,\tag{37}
$$

$$
\phi(\mathbf{x},t) = 2\left[\kappa\_1(t) - \eta\_1(t)\sqrt{(1-a^2)}\right]\mathbf{x}
$$

$$+2\int\_{0}^{t} D(\tau) \left[\kappa\_1^2 + \eta\_1^2 \left(3 - a^2\right) - 2\kappa\_1 \eta\_1 \sqrt{(1 - a^2)}\right] d\tau. \tag{38}$$

Dark soliton (36) has an additional parameter, 0 ≤ *a* ≤ 1, which designates the depth of modulation (the blackness of gray soliton) and its velocity against the background. When *a* = 1, dark soliton becomes black. For optical applications, Eq.(36) can be easily transformed into the Hasegawa and Tappert form for the nonautonomous dark solitons (Hasegawa, 1995) under the condition *κ*<sup>0</sup> = *η*<sup>0</sup> (<sup>1</sup> <sup>−</sup> *<sup>a</sup>*2) that corresponds to the special choice of the retarded frame associated with the group velocity of the soliton

$$q\_1^-(\mathbf{x}, t \mid \sigma = -1) = 2\eta\_1(t)\sqrt{\frac{D(t)}{R(t)}} \left[\sqrt{(1 - a^2)} + ia \tanh \tilde{\xi}\left(\mathbf{x}, t\right)\right]$$

$$\times \exp\left\{-i\left(\frac{\Theta(t)}{2}\mathbf{x}^2 + \tilde{\Phi}(\mathbf{x}, t)\right)\right\},$$

$$\tilde{\zeta}(\mathbf{x}, t) = 2a\eta\_1(t)\mathbf{x} + 4a\int\_0^t D(\tau)\eta\_1(\tau)\left[\eta\_1(\tau)\sqrt{(1 - a^2)} + K(\tau)\right]d\tau,$$

$$\tilde{\phi}(\mathbf{x}, t) = 2K(t)\mathbf{x} + 2\int\_0^t D(\tau)\left[K^2(\tau) + 2\eta\_1^2(\tau)\right]d\tau,$$

$$K(t) = \frac{R(t)}{D(t)}\int\_0^t \lambda\_0(\tau)\frac{D(\tau)}{R(\tau)}d\tau.$$

Notice that the solutions considered here hold only when the nonlinearity, dispersion and confining harmonic potential are related by Eq. (31), and both *D*(*t*) �= 0 and *R*(*t*) �= 0 for all times by definition.

Two-soliton *q*2(*x*, *t*) solution for *σ* = +1 follows from Eq. (34)

$$q\_2(\mathbf{x}, t) = 4 \sqrt{\frac{D(t)}{R(t)}} \frac{\mathbf{N} \begin{pmatrix} \mathbf{x}, t \\ \mathbf{D} \end{pmatrix}}{\mathbf{D} \begin{pmatrix} \mathbf{x}, t \end{pmatrix}} \exp\left[-\frac{i}{2} \Theta(t) \mathbf{x}^2\right],\tag{39}$$

where the numerator N (*x*, *t*) is given by

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

sech [*ξ*1(*x*, *t*)] × exp

 ,

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

<sup>2</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*(*x*, *<sup>t</sup>*)

(<sup>1</sup> − *<sup>a</sup>*2)

Dark soliton (36) has an additional parameter, 0 ≤ *a* ≤ 1, which designates the depth of modulation (the blackness of gray soliton) and its velocity against the background. When *a* = 1, dark soliton becomes black. For optical applications, Eq.(36) can be easily transformed into the Hasegawa and Tappert form for the nonautonomous dark solitons (Hasegawa, 1995)

 *x*

<sup>2</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*(*x*, *<sup>t</sup>*)

 *η*1(*τ*) 

*K*2(*τ*) + 2*η*<sup>2</sup>

*λ*0(*τ*)

<sup>1</sup> (*τ*) *dτ*,

*D*(*τ*) *R*(*τ*) *dτ*.

− 2*κ*1*η*<sup>1</sup>

(<sup>1</sup> <sup>−</sup> *<sup>a</sup>*2) that corresponds to the special choice of the retarded

(<sup>1</sup> − *<sup>a</sup>*2) + *ia* tanh *<sup>ζ</sup>*

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

 ,

Θ(*t*)

 *t*

0

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

Θ(*t*)

*D*(*τ*)*η*1(*τ*)

 *t*

0

Notice that the solutions considered here hold only when the nonlinearity, dispersion and confining harmonic potential are related by Eq. (31), and both *D*(*t*) �= 0 and *R*(*t*) �= 0 for all

*κ*1(*t*) − *η*1(*t*)

*D*(*τ*) *κ*2 <sup>1</sup> <sup>+</sup> *<sup>η</sup>*<sup>2</sup> 1 <sup>3</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 

<sup>1</sup> (*x*, *t*) soliton solutions are represented by the following analytic

 −*i* Θ(*t*)

(<sup>1</sup> − *<sup>a</sup>*2) + *ia* tanh *<sup>ζ</sup>* (*x*, *<sup>t</sup>*)

*D*(*τ*)*η*1(*τ*)*κ*1(*τ*)*dτ*, (37)

(<sup>1</sup> − *<sup>a</sup>*2)

(*x*, *t*) 

> *dτ*,

<sup>2</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>χ</sup>*1(*x*, *<sup>t</sup>*)

*dτ*. (38)

; (35)

(36)

Bright *q*<sup>+</sup>

*q*+

expressions:

<sup>1</sup> (*x*, *t*) and dark *q*<sup>−</sup>

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

<sup>1</sup> (*x*, *t* | *σ* = −1) = 2*η*1(*t*)

 −*i*

× exp

*ζ*(*x*, *t*) = 2*aη*1(*t*)*x* + 4*a*

+2 *t*

frame associated with the group velocity of the soliton

<sup>1</sup> (*x*, *t* | *σ* = −1) = 2*η*1(*t*)

(*x*, *t*) = 2*aη*1(*t*)*x* + 4*a*

*φ*(*x*, *t*) = 2*K*(*t*)*x* + 2

× exp

 −*i*

> *t*

0

*D*(*τ*) 

*<sup>K</sup>*(*t*) = *<sup>R</sup>*(*t*) *D*(*t*)

 *t*

0

0

*φ*(*x*, *t*) = 2

under the condition *κ*<sup>0</sup> = *η*<sup>0</sup>

*q*−

*ζ*

times by definition.

<sup>1</sup> (*x*, *t* | *σ* = +1) = 2*η*1(*t*)

*q*−

$$\begin{aligned} \mathbf{N} &= \cosh \mathfrak{f}\_2 \exp \left( -i \chi\_1 \right) \\ &\times \left[ \left( \kappa\_2 - \kappa\_1 \right)^2 + 2i \eta\_2 \left( \kappa\_2 - \kappa\_1 \right) \tanh \mathfrak{f}\_2 + \eta\_1^2 - \eta\_2^2 \right] + \eta\_2 \cosh \mathfrak{f}\_1 \exp \left( -i \chi\_2 \right) \end{aligned}$$
 
$$\times \left[ \left( \kappa\_2 - \kappa\_1 \right)^2 - 2i \eta\_1 \left( \kappa\_2 - \kappa\_1 \right) \tanh \mathfrak{f}\_1 - \eta\_1^2 + \eta\_2^2 \right] . \tag{40}$$

and the denominator D (*x*, *t*) is represented by

$$\begin{aligned} \mathbf{D} &= \cosh(\mathfrak{f}\_1 + \mathfrak{f}\_2) \left[ \left( \mathfrak{x}\_2 - \mathfrak{x}\_1 \right)^2 + \left( \mathfrak{y}\_2 - \mathfrak{y}\_1 \right)^2 \right] \\ &+ \cosh(\mathfrak{f}\_1 - \mathfrak{f}\_2) \left[ \left( \mathfrak{x}\_2 - \mathfrak{x}\_1 \right)^2 + \left( \mathfrak{y}\_2 + \mathfrak{y}\_1 \right)^2 \right] - 4 \eta\_1 \eta\_2 \cos \left( \chi\_2 - \chi\_1 \right) . \end{aligned} \tag{41}$$

Arguments and phases in Eqs.(39-41)

$$\xi\_i(\mathbf{x}, t) = 2\eta\_i(t)\mathbf{x} + 4 \int\_0^t D(\tau)\eta\_i(\tau)\kappa\_i(\tau)d\tau,\tag{42}$$

$$\chi\_i(\mathbf{x}, t) = 2\kappa\_i(t)\mathbf{x} + 2\int\_0^t D(\tau) \left[\kappa\_i^2(\tau) - \eta\_i^2(\tau)\right] d\tau \tag{43}$$

are related with the amplitudes

$$
\eta\_i(t) = \frac{D\_0 R(t)}{R\_0 D(t)} \eta\_{0i'} \tag{44}
$$

and velocities

$$\kappa\_i(t) = \frac{D\_0 \mathcal{R}(t)}{R\_0 D(t)} \left[ \kappa\_{0i} + \frac{R\_0}{D\_0} \int\_0^t \frac{\lambda\_0(\tau) D(\tau)}{R(\tau)} d\tau \right] \tag{45}$$

of the nonautonomous solitons, where *κ*0*<sup>i</sup>* and *η*0*<sup>i</sup>* correspond to the initial velocity and amplitude of the *i* -th soliton (*i* = 1, 2).

Eqs. (39-45) describe the dynamics of two bounded solitons at all times and all locations. Obviously, these soliton solutions reduce to classical soliton solutions in the limit of autonomous nonlinear and dispersive systems given by conditions: *R*(*t*) = *D*(*t*) = 1, and *λ*0(*t*) = Ω(*t*) ≡ 0 for canonical NLSE without external potentials.

if dispersion and nonlinearity evolve in unison *D*(*t*) = *R*(*t*) or *D* = *R* = 1, the solitons

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 63

Consider in more details the case when the nonlinearity *R* = *R*<sup>0</sup> stays constant but the

*D*(*Z*) = *D*<sup>0</sup> exp (−*c*0*Z*), Θ(*Z*) = Θ<sup>0</sup> exp (*c*0*Z*).

Let us write the one and two soliton solutions in this case with the lineal potential that, for

N(*Z*, *T*) <sup>D</sup>(*Z*, *<sup>T</sup>*) exp

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

[exp (*c*0*Z*) − 1] + 4*D*0*κ*0*<sup>i</sup>*

The initial velocity and amplitude of the *i* -th soliton (*i* = 1, 2) are denoted by *κ*0*<sup>i</sup>* and *η*0*i*. We display in Fig.1(a,b) the main features of nonautonomous colored solitons to show not only their acceleration and reflection from the lineal potential, but also their compression and amplitude amplification. Dark soliton propagation and dynamics are presented in Fig.1(c,d). The limit case of the Eqs.(48-51) appears when *c*<sup>0</sup> → ∞ (that means *D*(*Z*) = *D*<sup>0</sup> =constant) and corresponds to the Chen and Liu model (Chen, 1976). The solitons with argument and

*<sup>T</sup>* <sup>+</sup> <sup>2</sup>*κ*0*<sup>Z</sup>* <sup>+</sup> *<sup>α</sup>*0*Z*<sup>2</sup> <sup>−</sup> *<sup>T</sup>*<sup>0</sup>

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

represents the particle-like solutions which may be accelerated and reflected from the lineal

*α*0 *c*0 

exp (*c*0*Z*) − exp (−*c*0*Z*) *c*0

> ,

> > *Z* + 2*κ*0*α*0*Z*<sup>2</sup> +

2 3 *α*2 0*Z*<sup>3</sup>

where the nominator N(*Z*, *T*) and denominator D(*Z*, *T*) are given by Eqs. (40,41) and

[exp (*c*0*Z*) − 1] +

− *i* 2

> − *i* 2

*α*0 *c*0 

exp (*c*0*Z*) − 1 *c*0

 exp (2*c*0*Z*) − 1 2*c*<sup>0</sup>

> exp (*c*0*Z*) − 1 *c*0

> > − 2*Z*

<sup>Θ</sup><sup>0</sup> exp (*c*0*Z*) *<sup>T</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>χ*1(*Z*, *<sup>T</sup>*)

− *Z*

− *t* 

. (51)

Θ<sup>0</sup> exp (*c*0*Z*) *T*<sup>2</sup>

 , (48)

, (49)

, (50)

propagate with identical spectra, but with totally different time-space behavior.

*<sup>D</sup>*<sup>0</sup> exp (*c*0*Z*)sech [*ξ*1(*Z*, *<sup>T</sup>*)] <sup>×</sup> exp

*D*<sup>0</sup> exp (−*c*0*Z*)

*ξi*(*Z*, *T*) = 2*η*0*iT* exp (*c*0*Z*) + 4*D*0*η*0*<sup>i</sup>*

dispersion varies exponentially along the propagation distance

simplicity, does not depend on time: *λ*0(*Z*) = *α*<sup>0</sup> = *const*

× *κ*0*<sup>i</sup> c*0

*χi*(*Z*, *T*) = 2*κ*0*iT* exp (*c*0*Z*) + 2*D*<sup>0</sup>

<sup>+</sup>2*<sup>T</sup> <sup>α</sup>*<sup>0</sup> *c*0

+2*D*<sup>0</sup>

*ξ*(*Z*, *T*) = 2*η*<sup>0</sup>

 *α*<sup>0</sup> *c*0

*χ*(*Z*, *T*) = 2*κ*0*T* + 2*α*0*TZ* + 2

<sup>2</sup>

*U*2(*Z*, *T*) = 4

*<sup>U</sup>*1(*Z*, *<sup>T</sup>*) = <sup>2</sup>*η*<sup>01</sup>

phase

potential.
