Optical solitons of the resonant nonlinear Schrödinger equation with arbitrary index

Abstract

Traveling wave reduction of the resonant nonlinear Schrödinger equation with arbitrary refractive index is considered. The system of equations for real and imaginary parts is presented. The first integrals for the system of equations are found. This system of equations is transformed to the only governing nonlinear ordinary differential equations of the first order. Exact solutions in the form of periodic and solitary waves of this nonlinear ordinary differential equation are given for various values of the refractive index.

Introduction

At present, there is a great interest in the study of generalized nonlinear Schrödinger equations, in which resonant phenomenon are taken into account (see, for a example, [1], [2], [3], [4]). The self-similar propagation of light waves inside optical fibers with varying resonant and dispersion is considered in paper [1]. As this takes place a generalized resonant nonlinear Schrödinger equation and the distributed parameters has been used for description of the pulse evolution. The existence of nonlinear resonant states in a higher-order nonlinear Schrödinger model for describing the wave propagation in fiber optics, under certain parametric regime were studied in [2]. In paper [3] conservation laws and exact analytical solutions of resonant nonlinear Schrödinger’s equation with parabolic nonlinearity have been discussed and the modified Kudryashov algorithm for finding the exact solutions was applied. The resonant nonlinear Schrödinger’s equation has been studied with four forms of nonlinearity in [4] with application of the simplest equation method for solving the governing equations. Some other problems of propagation of nonlinear waves in a resonant medium described by the generalized non-linear Schrödinger equation have been also discussed in the works [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. In most of the works listed above, studies of generalized nonlinear Schrodinger equations that take into account resonant phenomena were carried out with specific indicators of the reflection coefficient.

However some recent papers, in particular [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], the attempts have been taken into account arbitrary power of nonlinearities in a number of generalizations for the nonlinear Schrodinger equation. In this paper, we are doing an attempt to transfer the approach developed in works [31], [32], [33], [34], [35], [36], [37], [38] to the resonant nonlinear Schrodinger equation.

In this paper we consider the generalization of the resonant nonlinear Schrödinger equation in the form

$$i{q}_t+a{q}_{xx}+\left(\alpha {\left|q\right|}^{2n}+\beta {\left|q\right|}^{4n}\right)q+\delta \left(\frac{{\left|q\right|}_{xx}}{\left|q\right|}\right)q-\epsilon q=0,$$

where $q(x,t)$ is the coplex-valued function, $i^2=-1$, $n\ne 0$, $a$, $\alpha$, $\beta$, $\delta$ and $\epsilon$ are parameters of Eq. (1), $x$ is coordinate and $t$ is time.

It is obviously that Eq. (1) is non-integrable partial differential equation. The solution for this equation cannot be found by the inverse scattering transform. At $n=1$ Eq. (1) has been studied earlier using some special methods popular recently for constructing non-integrable equations (see, for a example, [39], [40], [41], [42], [43], [44]).

The purpose of this work is to find general solution of Eq. (1) using the traveling wave reduction.