Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

Abstract

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with $p$-power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with $p=1,2,3,4$. No solutions for $p>1$ were known previously in the literature. For $p=1$, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) $p$ symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations  as well as to nonlinear ODE systems of generalized Hénon–Heiles form.