**6. Bound states of colored nonautonomous optical solitons: nonautonomous "agitated" breathers.**

Let us now give the explicit formula of the soliton solutions (48,49) for the case where all eigenvalues are pure imaginary, or the initial velocities of the solitons are equal to zero. In the case *N* = 1 and *λ*0(*Z*) = 0 , we obtain

$$\begin{split} \mathcal{U}\_{1}(\boldsymbol{Z},\boldsymbol{T}) &= 2\eta\_{01}\sqrt{D\_{0}\exp\left(\boldsymbol{c}\_{0}\boldsymbol{Z}\right)}\mathrm{sech}\left[2\eta\_{01}\boldsymbol{T}\exp\left(\boldsymbol{c}\_{0}\boldsymbol{Z}\right)\right] \\ &\times \exp\left[-\frac{i}{2}\Theta\_{0}\exp\left(\boldsymbol{c}\_{0}\boldsymbol{Z}\right)\boldsymbol{T}^{2} + i\mathcal{D}\_{0}\eta\_{01}^{2}\frac{\exp\left(2\boldsymbol{c}\_{0}\boldsymbol{Z}\right) - 1}{2\boldsymbol{c}\_{0}}\right]. \end{split} \tag{52}$$

This result shows that the laws of soliton adaptation to the external potentials (31) allow to stabilize the soliton even without a trapping potential. In addition, Eq.(52) indicates the possibility for the optimal compression of solitons, which is shown in Fig.2. We stress that direct computer experiment confirms the exponential in time soliton compression scenario in full accordance with analytical expression Eq.(52).

The bound two-soliton solution for the case of the pure imaginary eigenvalues is represented by

$$\mathrm{LI}\_{2}(Z,T) = 4\sqrt{D\_{0}\exp\left(-c\_{0}Z\right)}\frac{\mathrm{N}\left(Z,T\right)}{\mathrm{D}\left(Z,T\right)}\exp\left[-\frac{i}{2}\Theta\_{0}\exp\left(c\_{0}Z\right)T^{2}\right],\tag{53}$$

where

$$\mathcal{N} = \left(\eta\_{01}^2 - \eta\_{02}^2\right) \exp\left(c\_0 Z\right) \left[\eta\_{01} \cosh\xi\_2 \exp\left(-i\chi\_1\right) - \eta\_{02} \cosh\xi\_1 \exp\left(-i\chi\_2\right)\right],\tag{54}$$

$$\mathbf{D} = \cosh(\mathfrak{f}\_1 + \mathfrak{f}\_2) \left(\eta\_{01} - \eta\_{02}\right)^2 + \cosh(\mathfrak{f}\_1 - \mathfrak{f}\_2) \left(\eta\_{01} + \eta\_{02}\right)^2 - 4\eta\_{01}\eta\_{02}\cos\left(\chi\_2 - \chi\_1\right), \quad \text{(55)}$$

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

and

14 Will-be-set-by-IN-TECH

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)

**6. Bound states of colored nonautonomous optical solitons: nonautonomous**

Let us now give the explicit formula of the soliton solutions (48,49) for the case where all eigenvalues are pure imaginary, or the initial velocities of the solitons are equal to zero. In the

*D*<sup>0</sup> exp (*c*0*Z*)sech [2*η*01*T* exp (*c*0*Z*))]

This result shows that the laws of soliton adaptation to the external potentials (31) allow to stabilize the soliton even without a trapping potential. In addition, Eq.(52) indicates the possibility for the optimal compression of solitons, which is shown in Fig.2. We stress that direct computer experiment confirms the exponential in time soliton compression scenario in

The bound two-soliton solution for the case of the pure imaginary eigenvalues is represented

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

<sup>2</sup> <sup>+</sup> cosh(*ξ*<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*2)(*η*<sup>01</sup> <sup>+</sup> *<sup>η</sup>*02)

 − *i* 2

exp (*c*0*Z*) [*η*<sup>01</sup> cosh *ξ*<sup>2</sup> exp (−*iχ*1) − *η*<sup>02</sup> cosh *ξ*<sup>1</sup> exp (−*iχ*2)] , (54)

Θ<sup>0</sup> exp (*c*0*Z*) *T*<sup>2</sup> + *i*2*D*0*η*<sup>2</sup>

01

exp (2*c*0*Z*) − 1 2*c*<sup>0</sup>

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

<sup>2</sup> <sup>−</sup> <sup>4</sup>*η*01*η*<sup>02</sup> cos (*χ*<sup>2</sup> <sup>−</sup> *<sup>χ</sup>*1), (55)

. (52)

, (53)

R=–D=1.0 and *α*<sup>0</sup> = −1.0 and (d) R=–D=cos( *ωZ*), where *ω* = 3.0.

× exp − *i* 2

full accordance with analytical expression Eq.(52).

*D*<sup>0</sup> exp (−*c*0*Z*)

**"agitated" breathers.**

by

where

N = *η*2 <sup>01</sup> <sup>−</sup> *<sup>η</sup>*<sup>2</sup> 02 

case *N* = 1 and *λ*0(*Z*) = 0 , we obtain

*U*1(*Z*, *T*) = 2*η*<sup>01</sup>

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

D = cosh(*ξ*<sup>1</sup> + *ξ*2)(*η*<sup>01</sup> − *η*02)

$$\xi\_i(Z, T) = 2\eta\_{0i} T \exp\left(\mathbf{c}\_0 Z\right),\tag{56}$$

$$\chi\_i(Z, T) = -2D\_0 \eta\_{0i}^2 \frac{\exp\left(2c\_0 Z\right) - 1}{2c\_0} + \chi\_{i0}.\tag{57}$$

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

$$\mathcal{U}\_2(Z, T) = 4\sqrt{D\_0 \exp\left(-c\_0 Z\right)} \exp\left[-\frac{i}{2} \Theta\_0 \exp\left(c\_0 Z\right) T^2\right] \tag{58}$$

$$\begin{split} \times & \exp\left[\frac{i}{4c\_0} D\_0 \left[\exp\left(2c\_0 Z\right) - 1\right] + \chi\_{10}\right] \\ & \times \frac{\cosh 3X - 3\cosh X \exp\left\{i2D\_0 \left[\exp\left(2c\_0 Z\right) - 1\right]/c\_0 + i\Delta \theta\right\}}{\cosh 4X + 4\cosh 2X - 3\cos\left\{2D\_0 \left[\exp\left(2c\_0 Z\right) - 1\right]/c\_0 + \Delta \theta\right\}} \end{split} \tag{59}$$

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

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 as the Satsuma-Yajima breather:

$$\mathcal{U}l\_2(Z,T) = 4 \frac{\cosh 3T + 3 \cosh T \exp\left(4iZ\right)}{\cosh 4T + 4 \cosh 2T + 3 \cos 4Z} \exp\left(\frac{iZ}{2}\right).$$

At *Z* = 0 it takes the simple form *U*(*Z*, *T*) = 2*sech*(*T*). An interesting property of this solution is that its form oscillates with the so-called soliton period *Tsol* = *π*/2.

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 nonautonomous system.

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.

and five high-intensity and "long-lived" components.

ocean.

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

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 67

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

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