Variational problems for the system of nonlinear Schr\"odinger equations with derivative nonlinearities

TL;DR

This paper investigates the existence, stability, and well-posedness of solutions for a system of nonlinear Schrödinger equations with derivative nonlinearities, motivated by laser-plasma interaction models, using variational methods.

Contribution

It provides new results on ground state existence, stability of traveling waves, and global well-posedness for the system with derivative nonlinearities, addressing open problems posed by Colin-Colin.

Findings

Existence of ground state solutions established.

Stability of traveling waves with small speed proved in 1D.

Global well-posedness demonstrated for the system.

Abstract

We consider the Cauchy problem of the system of nonlinear Schr\"odinger equations with derivative nonlinearlity. This system was introduced by Colin-Colin (2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin-Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for $1$-dimension.