Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

Abstract

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.