**5. Chirped optical solitons with moving spectra in nonautonomous systems: colored nonautonomous solitons**

Both the nonlinear Schrödinger equations (28, 30) and the Lax pair equations (3–6) are written down here in the most general form. The transition to the problems of optical solitons is accomplished by the substitution *<sup>x</sup>* <sup>→</sup> *<sup>T</sup>* (or *<sup>x</sup>* <sup>→</sup> *<sup>X</sup>*); *<sup>t</sup>* <sup>→</sup> *<sup>Z</sup>* and *<sup>q</sup>*+(*x*, *<sup>t</sup>*) <sup>→</sup> *<sup>u</sup>*�+(*Z*, *<sup>T</sup>*( *or X*)) for bright solitons, and � *q*−(*x*, *t*) �<sup>∗</sup> <sup>→</sup> *<sup>u</sup>*�−(*Z*, *<sup>T</sup>*( *or X*)) for dark solitons, where the asterisk denotes the complex conjugate, *Z* is the normalized distance, and *T* is the retarded time for temporal solitons, while *X* is the transverse coordinate for spatial solitons.

The important special case of Eq.(30) arises under the condition Ω2(*Z*) = 0. Let us rewrite Eq. (30) by using the reduction Ω = 0, which denotes that the confining harmonic potential is vanishing

$$\left(i\frac{\partial u}{\partial Z} + \frac{\sigma}{2}D(Z)\frac{\partial^2 u}{\partial T^2} + R(Z)\left|u\right|^2 u - 2\sigma\lambda\_0(Z)Tu = 0. \tag{46}$$

This implies that the self-induced soliton phase shift Θ(*Z*), dispersion *D*(*Z*), and nonlinearity *R*(*Z*) are related by the following law of soliton adaptation to external linear potential

$$D(Z)/D\_0 = R(Z)/R\_0 \exp\left\{-\frac{\Theta\_0 D\_0}{R\_0} \int\_0^Z R(\tau)d\tau\right\}.\tag{47}$$

Nonautonomous exactly integrable NLSE model given by Eqs. (46,47) can be considered as the generalization of the well-studied Chen and Liu model (Chen, 1976) with linear potential *λ*0(*Z*) ≡ *α*<sup>0</sup> = *const* and *D*(*Z*) = *D*<sup>0</sup> = *R*(*Z*) = *R*<sup>0</sup> = 1, *σ* = +1, Θ<sup>0</sup> = 0. It is interesting to note that the accelerated solitons predicted by Chen and Liu in plasma have been discovered in nonlinear fiber optics only decade later (Agrawal, 2001; Dianov et al., 1989; Taylor, 1992). Notice that nonautonomous solitons with nontrivial self-induced phase shifts and varying amplitudes, speeds and spectra for Eq. (46) are given in quadratures by Eqs. (35-45) under condition Ω2(*Z*) = 0.

Let us show that the so-called Raman colored optical solitons can be approximated by this equation. Self-induced Raman effect (also called as soliton self-frequency shift) is being described by an additional term in the NLSE: −*σRU∂* | *U* | <sup>2</sup> /*∂T*, where *σ<sup>R</sup>* originates from the frequency dependent Raman gain (Agrawal, 2001; Dianov et al., 1989; Taylor, 1992). Assuming that soliton amplitude does not vary significantly during self-scattering | *U* | <sup>2</sup>= *η*2sech2(*ηT*), we obtain that

$$
\sigma\_R \frac{\partial \mid \mathcal{U} \mid^2}{\partial T} \approx -2\sigma\_R \eta^4 T = 2\alpha\_0 T
$$

and *dv*/*dZ* = *σRη*4/2, where *v* = *κ*/2. The result of soliton perturbation theory (Agrawal, 2001; Dianov et al., 1989; Taylor, 1992) gives *dv*/*dZ* = 8*σRη*4/15. This fact explains the remarkable stability of colored Raman solitons that is guaranteed by the property of the exact integrability of the Chen and Liu model (Chen, 1976). More general model Eq. (46) and its exact soliton solutions open the possibility of designing an effective soliton compressor, for example, by drawing a fiber with *R*(*Z*) = 1 and *D*(*Z*) = exp(−*c*0*Z*),where *c*<sup>0</sup> = Θ0*D*0. It seems very attractive to use the results of nonautonomous solitons concept in ultrashort photonic applications and soliton lasers design.

Another interesting feature of the novel solitons, which we called colored nonautonomous solitons, is associated with the nontrivial dynamics of their spectra. Frequency spectrum of the chirped nonautonomous optical soliton moves in the frequency domain. In particular, 12 Will-be-set-by-IN-TECH

Both the nonlinear Schrödinger equations (28, 30) and the Lax pair equations (3–6) are written down here in the most general form. The transition to the problems of optical solitons is accomplished by the substitution *<sup>x</sup>* <sup>→</sup> *<sup>T</sup>* (or *<sup>x</sup>* <sup>→</sup> *<sup>X</sup>*); *<sup>t</sup>* <sup>→</sup> *<sup>Z</sup>* and *<sup>q</sup>*+(*x*, *<sup>t</sup>*) <sup>→</sup> *<sup>u</sup>*�+(*Z*, *<sup>T</sup>*( *or X*))

denotes the complex conjugate, *Z* is the normalized distance, and *T* is the retarded time for

The important special case of Eq.(30) arises under the condition Ω2(*Z*) = 0. Let us rewrite Eq. (30) by using the reduction Ω = 0, which denotes that the confining harmonic potential is

This implies that the self-induced soliton phase shift Θ(*Z*), dispersion *D*(*Z*), and nonlinearity *R*(*Z*) are related by the following law of soliton adaptation to external linear potential

Nonautonomous exactly integrable NLSE model given by Eqs. (46,47) can be considered as the generalization of the well-studied Chen and Liu model (Chen, 1976) with linear potential *λ*0(*Z*) ≡ *α*<sup>0</sup> = *const* and *D*(*Z*) = *D*<sup>0</sup> = *R*(*Z*) = *R*<sup>0</sup> = 1, *σ* = +1, Θ<sup>0</sup> = 0. It is interesting to note that the accelerated solitons predicted by Chen and Liu in plasma have been discovered in nonlinear fiber optics only decade later (Agrawal, 2001; Dianov et al., 1989; Taylor, 1992). Notice that nonautonomous solitons with nontrivial self-induced phase shifts and varying amplitudes, speeds and spectra for Eq. (46) are given in quadratures by Eqs. (35-45) under

Let us show that the so-called Raman colored optical solitons can be approximated by this equation. Self-induced Raman effect (also called as soliton self-frequency shift) is being

frequency dependent Raman gain (Agrawal, 2001; Dianov et al., 1989; Taylor, 1992). Assuming

and *dv*/*dZ* = *σRη*4/2, where *v* = *κ*/2. The result of soliton perturbation theory (Agrawal, 2001; Dianov et al., 1989; Taylor, 1992) gives *dv*/*dZ* = 8*σRη*4/15. This fact explains the remarkable stability of colored Raman solitons that is guaranteed by the property of the exact integrability of the Chen and Liu model (Chen, 1976). More general model Eq. (46) and its exact soliton solutions open the possibility of designing an effective soliton compressor, for example, by drawing a fiber with *R*(*Z*) = 1 and *D*(*Z*) = exp(−*c*0*Z*),where *c*<sup>0</sup> = Θ0*D*0. It seems very attractive to use the results of nonautonomous solitons concept in ultrashort

Another interesting feature of the novel solitons, which we called colored nonautonomous solitons, is associated with the nontrivial dynamics of their spectra. Frequency spectrum of the chirped nonautonomous optical soliton moves in the frequency domain. In particular,

*<sup>∂</sup><sup>T</sup>* ≈ −2*σRη*4*<sup>T</sup>* <sup>=</sup> <sup>2</sup>*α*0*<sup>T</sup>*

that soliton amplitude does not vary significantly during self-scattering | *U* |

*∂* | *U* | 2 ⎧ ⎨ ⎩<sup>−</sup> <sup>Θ</sup>0*D*<sup>0</sup> *R*0

*<sup>∂</sup>T*<sup>2</sup> <sup>+</sup> *<sup>R</sup>*(*Z*)|*u*<sup>|</sup>

�<sup>∗</sup> <sup>→</sup> *<sup>u</sup>*�−(*Z*, *<sup>T</sup>*( *or X*)) for dark solitons, where the asterisk

� *Z*

*R*(*τ*)*dτ*

0

<sup>2</sup> *<sup>u</sup>* <sup>−</sup> <sup>2</sup>*σλ*0(*Z*)*Tu* <sup>=</sup> 0. (46)

<sup>⎭</sup> . (47)

<sup>2</sup> /*∂T*, where *σ<sup>R</sup>* originates from the

<sup>2</sup>= *η*2sech2(*ηT*),

⎫ ⎬

**5. Chirped optical solitons with moving spectra in nonautonomous systems:**

**colored nonautonomous solitons**

*i ∂u <sup>∂</sup><sup>Z</sup>* <sup>+</sup> *σ* <sup>2</sup> *<sup>D</sup>*(*Z*)

*q*−(*x*, *t*)

temporal solitons, while *X* is the transverse coordinate for spatial solitons.

*∂*2*u*

*D*(*Z*)/*D*<sup>0</sup> = *R*(*Z*)/*R*<sup>0</sup> exp

described by an additional term in the NLSE: −*σRU∂* | *U* |

photonic applications and soliton lasers design.

*σR*

for bright solitons, and �

condition Ω2(*Z*) = 0.

we obtain that

vanishing

if dispersion and nonlinearity evolve in unison *D*(*t*) = *R*(*t*) or *D* = *R* = 1, the solitons propagate with identical spectra, but with totally different time-space behavior.

Consider in more details the case when the nonlinearity *R* = *R*<sup>0</sup> stays constant but the dispersion varies exponentially along the propagation distance

$$\begin{aligned} D(Z) &= D\_0 \exp\left(-c\_0 Z\right), \\ \Theta(Z) &= \Theta\_0 \exp\left(c\_0 Z\right). \end{aligned}$$

Let us write the one and two soliton solutions in this case with the lineal potential that, for simplicity, does not depend on time: *λ*0(*Z*) = *α*<sup>0</sup> = *const*

$$\mathcal{U}\_1(Z,T) = 2\eta\_0 \sqrt{D\_0 \exp\left(c\_0 Z\right)} \text{sech}\left[\xi\_1(Z,T)\right] \times \exp\left[-\frac{i}{2} \Theta\_0 \exp\left(c\_0 Z\right)T^2 - i\chi\_1(Z,T)\right],\tag{48}$$

$$\mathrm{U}\_{2}(Z,T) = 4\sqrt{D\_{0}\exp\left(-c\_{0}Z\right)}\frac{\mathrm{N}(Z,T)}{\mathrm{D}(Z,T)}\exp\left[-\frac{i}{2}\Theta\_{0}\exp\left(c\_{0}Z\right)T^{2}\right],\tag{49}$$

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

$$\begin{aligned} \tilde{\xi}\_i(Z, T) &= 2\eta\_{0i} T \exp\left(c\_0 Z\right) + 4D\_0 \eta\_{0i} \\\\ &\times \left\{ \frac{\kappa\_{0i}}{c\_0} \left[ \exp\left(c\_0 Z\right) - 1 \right] + \frac{\kappa\_0}{c\_0} \left[ \frac{\exp\left(c\_0 Z\right) - 1}{c\_0} - Z \right] \right\}, \end{aligned} \tag{50}$$

$$\begin{aligned} \chi\_i(Z, T) &= 2\kappa\_{0i} T \exp\left(c\_0 Z\right) + 2D\_0 \left(\kappa\_{0i}^2 - \eta\_{0i}^2\right) \frac{\exp\left(2c\_0 Z\right) - 1}{2c\_0} \\\\ &\times \quad \qquad \qquad \qquad \eta\_0 \left[ \exp\left(c\_0 Z\right) - 1 \right] \end{aligned}$$

$$+2T\frac{a\_0}{c\_0}\left[\exp\left(c\_0Z\right)-1\right]+4D\_0\kappa\_{0i}\frac{a\_0}{c\_0}\left[\frac{\exp\left(c\_0Z\right)-1}{c\_0}-t\right]$$

$$+2D\_0\left(\frac{a\_0}{c\_0}\right)^2\left[\frac{\exp\left(c\_0Z\right)-\exp\left(-c\_0Z\right)}{c\_0}-2Z\right].\tag{51}$$

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 phase

$$\begin{aligned} \xi(Z,T) &= 2\eta\_0 \left( T + 2\kappa\_0 Z + \kappa\_0 Z^2 - T\_0 \right), \\ \chi(Z,T) &= 2\kappa\_0 T + 2\kappa\_0 T Z + 2\left(\kappa\_0^2 - \eta\_0^2\right) Z + 2\kappa\_0 \kappa\_0 Z^2 + \frac{2}{3} a\_0^2 Z^3. \end{aligned}$$

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

Fig. 2. Self-compression of nonautonomous soliton calculated within the framework of the model Eq. (46) after choosing the soliton management parameters *c*<sup>0</sup> = 0.05; *α* = 0 and

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 65

0*i*

− *i* 2

*<sup>D</sup>*<sup>0</sup> [exp (2*c*0*Z*) <sup>−</sup> <sup>1</sup>] <sup>+</sup> *<sup>χ</sup>*<sup>10</sup>

In the *D*(*Z*) = *D*<sup>0</sup> = 1, *c*<sup>0</sup> = 0 limit, this solution is reduced to the well-known breather solution, which was found by Satsuma and Yajima (Satsuma & Yajima, 1974) and was called

At *Z* = 0 it takes the simple form *U*(*Z*, *T*) = 2*sech*(*T*). An interesting property of this solution

In more general case of the varying dispersion, *D*(*Z*) = *D*<sup>0</sup> exp (−*c*0*Z*), shown in Fig.3 (*c*<sup>0</sup> = 0.25, *η*<sup>10</sup> = 0.25, *η*<sup>20</sup> = 0.75), the soliton period, according to Eq.(58), depends on time. The Satsuma and Yajima breather solution can be obtained from the general solution if and only if the soliton phases are chosen properly, precisely when Δ*ϕ* = *π*. The intensity profiles of the wave build up a complex landscape of peaks and valleys and reach their peaks at the points of the maximum. Decreasing group velocity dispersion (or increasing nonlinearity) stimulates the Satsuma-Yajima breather to accelerate its period of "breathing" and to increase its peak amplitudes of "breathing", that is why we call this effect as "agitated breather" in

*<sup>U</sup>*2(*Z*, *<sup>T</sup>*) = <sup>4</sup> cosh 3*<sup>T</sup>* <sup>+</sup> 3 cosh *<sup>T</sup>* exp (4*iZ*)

is that its form oscillates with the so-called soliton period *Tsol* = *π*/2.

exp (2*c*0*Z*) − 1 2*c*<sup>0</sup>

cosh 3*X* − 3 cosh *X* exp {*i*2*D*<sup>0</sup> [exp (2*c*0*Z*) − 1] /*c*<sup>0</sup> + *i*Δ*ϕ*} cosh 4*X* + 4 cosh 2*X* − 3 cos {2*D*<sup>0</sup> [exp (2*c*0*Z*) − 1] /*c*<sup>0</sup> + Δ*ϕ*}

cosh 4*<sup>T</sup>* <sup>+</sup> 4 cosh 2*<sup>T</sup>* <sup>+</sup> 3 cos 4*<sup>Z</sup>* exp *iZ*

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

*ξi*(*Z*, *T*) = 2*η*0*iT* exp (*c*0*Z*), (56)

2 .

+ *χi*0. (57)

(58)

,

*η*<sup>0</sup> = 0.5. (a) the temporal behavior; (b) the corresponding contour map.

*<sup>χ</sup>i*(*Z*, *<sup>T</sup>*) = <sup>−</sup>2*D*0*η*<sup>2</sup>

*<sup>D</sup>*<sup>0</sup> exp (−*c*0*Z*) exp

For the particular case of *η*<sup>10</sup> = 1/2, *η*<sup>20</sup> = 3/2 Eqs.(53-57) are transformed to

and

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

×

where *X* = *T* exp(*c*0*Z*), Δ*ϕ* = *χ*<sup>20</sup> − *χ*10.

as the Satsuma-Yajima breather:

nonautonomous system.

<sup>×</sup> exp *<sup>i</sup>*

4*c*<sup>0</sup>

Fig. 1. Evolution of nonautonomous bright (a,b) optical soliton calculated within the framework of the generalized model given by Eqs. (46-51) after choosing the soliton management parameters *c*0=0.05, *α*<sup>0</sup> = –0.2, *η*<sup>10</sup> = 0.5, *κ*<sup>10</sup> = 1.5. (a) the temporal behavior; (b) the corresponding contour map. (c,d) Dark nonautonomous soliton dynamics within the framework of the model Eqs. (46,47) after choosing the soliton management parameters: (c) R=–D=1.0 and *α*<sup>0</sup> = −1.0 and (d) R=–D=cos( *ωZ*), where *ω* = 3.0.
