On the existence of bound and ground states for some coupled nonlinear Schr\"odinger--Korteweg-de Vries equations

TL;DR

This paper proves the existence and classification of positive bound and ground states for coupled nonlinear Schrödinger--Korteweg-de Vries equations, extending previous results to more general systems and nonlinearities using variational methods.

Contribution

It introduces a novel variational approach to establish existence and multiplicity of solutions for coupled nonlinear Schrödinger--Korteweg-de Vries systems, including more general nonlinearities and multi-equation systems.

Findings

Existence of positive radially symmetric ground states under certain coupling conditions.

Existence of positive bound states for small coupling coefficients.

Extension of results to systems with more general power nonlinearities and multiple equations.

Abstract

We demonstrate existence of positive bound and ground states for a system of coupled nonlinear Schr\"odinger--Korteweg-de Vries equations. More precisely, we prove there is a positive radially symmetric ground state if either the coupling coefficient $\beta>\Lambda$ (for an appropriate constant $\Lambda>0$) or $\beta>0$ with appropriate conditions on the other parameters of the problem. Concerning bound states, we prove there exists a positive radially symmetric bound state if either $0<\beta$ is sufficiently small or $0<\beta<\Lambda$ with some appropriate conditions on the parameters as for the ground states. That results give a classification of positive solutions as well as multiplicity of positive solutions. Furthermore, we consider a system with more general power nonlinearities, proving the above results, and also we study natural extended systems with more than two equations. Although the techniques we employed are variational, we look for critical points of an energy functional different from the classical one used in this kind of systems. Our approach improves many of the previous known results, as well as permit us to show new results not considered or studied before.