Coherent structures and sequences of exact kink and anti-kink solutions to the complex cubic–quintic Ginzburg-Landau equation perturbed by intrapulse Raman scattering

Abstract

In this paper we have studied coherent structures composed of kink and anti-kink solutions in the complex cubic–quintic Ginzburg–Landau equation perturbed by intrapulse Raman scattering. We have formulated the required conditions for the parameters of the basic equation for the simultaneous existence of kink and anti-kink solutions. The performed numerical study has shown that such coherent structures preserve the temporal shape of the constituent solutions very well during their propagation. We have found that such structures could be created by means of proper super-Gaussian pulses, which means they represent some eigenmodes of the system. We have also found that sequences of such coherent structures can propagate over long distances without nonlinear interaction between them if the value of the ratio of the distance between them and the full-width half maximum of the structure is equal to two, i.e. in conditions where the classical Schrödinger soliton cannot spread without any interaction. In all the considered cases, even after a merging process has occurred of closely positioned structures, the coherent structures maintain their fronts in accordance with the exact kink and anti-kink solutions of the CCQGLE.

Appendix: Kink and anti-kink exact solutions of CCQGLE perturbed with IRS (Uzunov et al. 2023)

A detailed description of the procedure for the calculation of kink and anti-kink solutions to Eq. (1) is available in Refs. (Uzunov et al. 2023; Uzunov and Georgiev 2014). Here we have presented only the main results.

It is shown in Ref. (Uzunov and Georgiev 2014), that the complex partial differential Eq. (1) can be reduced to a system of ordinary differential equations through the ansatz

$$U\left( {x,t} \right) = u\left( \xi \right)\exp \left( {i\left( {f\left( \xi \right) + Kx} \right)} \right),$$

(A1)

where $\xi = t - Mx$, and where $M$ and $K$ are real numbers. The condition

$$v = \frac{\mu }{2\beta },\;\;M = \frac{{\left( {1 + 4\beta^{2} } \right)\left( {\varepsilon - 2\beta } \right)}}{{4\beta^{2} \gamma }}\,,\quad K = \frac{{\left( {1 + 4\beta^{2} } \right)\left( {\varepsilon - 2\beta } \right)^{2} }}{{16\beta^{4} \gamma^{2} }} - \frac{\delta }{2\beta }$$

(A2)

is valid (Uzunov and Georgiev 2014). $u\left( \xi \right)$ and $f\left( \xi \right)$ are real-value functions of the similar variable $\xi$ and they can be verified by direct computation, which, on its turn, and under specific conditions leads to a single differential equation of Liénard type for the amplitude function $u\left( \xi \right)$:

$$\,u^{\prime\prime} + \left( {c_{2} + c_{4} u^{2} } \right)u^{\prime} + c_{1} u + c_{3} u^{3} + c_{5} u^{5} = 0,$$

(A3)

where the magnitude of $c_{1} ,\,c_{2} ,\,c_{3} ,\,c_{4}$ and $c_{5}$ is given in Uzunov and Georgiev (2014):

$$\,c_{1} = \,\frac{\delta }{\beta } - \frac{{\left( {\varepsilon - 2\beta } \right)^{2} }}{{16\beta^{4} \gamma^{2} }},\,c_{2} = \,\frac{\varepsilon - 2\beta }{{\beta \gamma }},c_{3} = \frac{2 + 4\beta \varepsilon }{{1 + 4\beta^{2} }},c_{4} = - \frac{4\gamma }{{1 + 4\beta^{2} }},c_{5} = \frac{\mu }{\beta } - \frac{{4\beta^{2} \gamma^{2} }}{{\left( {1 + 4\beta^{2} } \right)^{2} }}.$$

(A4)

The conditions under which this equation is compatible with a sub-equation have been given in Ref. (Uzunov et al. 2023). By direct computation, we have found kink-like and anti-kink-like solutions of the form (A1) to equations of the form (1), which are given by using the following amplitude function (Uzunov et al. 2023):

$$\begin{aligned} u\left( \xi \right) & = \frac{1}{{\sqrt {v\left( \xi \right)} }}, \\ v\left( \xi \right) & = \frac{{c_{4}^{2} \pm c_{4} \sqrt {c_{4}^{2} - 12c_{5} } - 24c_{5} }}{{18c_{3} - 3c_{2} \left( {c_{4} \pm \sqrt {c_{4}^{2} - 12c_{5} } } \right)}} + C_{2} \exp \left[ {\frac{{\left( {c_{2} \sqrt {c_{4}^{2} - 12c_{5} } \pm c_{2} c_{4} \mp 6c_{3} } \right)}}{{2\sqrt {c_{4}^{2} - 12c_{5} } \mp c_{4} }}\left( {\xi + \xi_{0} } \right)} \right], \\ \end{aligned}$$

(A5)

where $C_{2}$ and $\xi_{0}$ are arbitrary constants. The first equation corresponds to the anti-kink solution whereas the second one: to the kink solution.