Stability of bound states for regularized nonlinear Schr\"odinger equations

TL;DR

This paper investigates the stability of bound states in regularized nonlinear Schrödinger equations, establishing conditions for existence and stability of ground states and demonstrating broader stability ranges compared to the classical case.

Contribution

It provides new sufficient conditions for the existence and orbital stability of ground states in regularized NLS equations, extending stability results beyond the classical nonregularized case.

Findings

Ground states exist and are stable over a wider range of nonlinearities.

Sufficient conditions for stability of general bound states are established.

Some stable bound states are not ground states.

Abstract

We consider the stability of bound-state solutions of a family of regularized nonlinear Schr\"odinger equations which were introduced by Dumas, Lannes and Szeftel as models for the propagation of laser beams. Among these bound-state solutions are ground states, which are defined as solutions of a variational problem. We give a sufficient condition for existence and orbital stability of ground states, and use it to verify that ground states exist and are stable over a wider range of nonlinearities than for the nonregularized nonlinear Schr\"odinger equation. We also give another sufficient and almost necessary condition for stability of general bound states, and show that some stable bound states exist which are not ground states.