Coupled system of nonlinear Schr\"odinger and Korteweg-de Vries equations

TL;DR

This paper investigates the existence and multiplicity of solutions for coupled nonlinear Schr"odinger and Korteweg-de Vries equations, including higher-order systems, using analytical methods for standing and traveling wave solutions.

Contribution

It extends previous results by analyzing higher-order coupled systems and establishing conditions for existence and multiplicity of solutions.

Findings

Existence of positive bound and ground states for second-order systems

Existence and multiplicity of solutions for higher-order bi-harmonic systems

Conditions on parameters for solution existence

Abstract

This work is divided into two parts. First, we analyze the existence of positive bound and ground states for a second order stationary system coming from a coupled system of nonlinear Schr\"odinger--Korteweg-de Vries equations. Second, we extend these results for a higher order system of nonlinear Schr\"odinger--Korteweg-de Vries equations. Looking for "standing-traveling" waves we arrive at a bi-harmonic stationary system, for which we prove the existence and multiplicity of solutions under appropriate conditions on the parameters.