Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index
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Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index Nikolay A. Kudryashov* National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, 115409 Moscow, Russia Received August 06, 2020; revised September 08, 2020; accepted September 22, 2020
Abstract—A nonlinear fourth-order differential equation with arbitrary refractive index for description of the pulse propagation in an optical fiber is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for solutions of the equation using the traveling wave reduction. We present a novel method for finding soliton solutions of nonlinear evolution equations. The essence of this method is based on the hypothesis about the possible type of an auxiliary equation with an already known solution. This new auxiliary equation is used as a basic equation to look for soliton solutions of the original equation. We have found three forms of soliton solutions of the equation at some constraints on parameters of the equation. MSC2010 numbers: 34M55 DOI: 10.1134/S1560354720060039 Keywords: nonlinear mathematical model, traveling wave, solitary wave, pulse propagation, optical fiber
1. INTRODUCTION Currently, we can observe a huge interest in the development of nonlinear mathematical models for describing the pulse propagation of various types in an optical fiber. Let us mention here the most popular mathematical models that have been intensively studied in the last few years: the Ginzburg – Landau equation [1–4], the Radhakrishnan – Kundu – Laksmanan equation [5–8], the nonlinear Schr¨ odinger equation with cubic-quintic nonlinear law [9–12], the Triki – Biswas equation [13–15], the Kundu – Mukherjee – Naskar model [16–19], the Fokas – Lenells equation [20– 23], the perturbed Chen – Lee – Liu model [24–27], the Biswas – Milovic equation [28–31], the perturbed Gerdjikov – Ivanov model [32–35], the Kudryashov equation [36–41], and the Biswas – Arshed equation [42–45]. For the transmission of information in optical lines, it is particularly important to take into account the influence of high-order dispersion, various reflection laws in the optical fiber, and processes of nonlocal nonlinearity [46–49]. In this paper we first consider the nonlinear fourth-order differential equation in the form i qt + α qxx + i β qxxx + δ qxxxx + h (|q|n )xx + g (|q|2n )xx q (1.1) + a |q|n + b |q|2n + c |q|3n + d |q|4n q = 0, where n is refractive index and α, β, δ, a, b, c, d, h, g are dimensional parameters of the mathematical model. The monomial with the fourth derivative in Eq. (1.1) takes into account highorder dispersion processes, and nonlinear monomials with the second derivative are responsible for nonlocal nonlinearity processes, and the last four expressions are a characteristic of the potential for pulse propagation in an optical fiber. *
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Equation (1.1) does not pass the Painlev´e test and the Cauchy problem for this equation c
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