Spectral stability of periodic NLS and CGL solutions

TL;DR

This paper investigates the spectral stability of periodic traveling wave solutions in the focusing NLS and their persistence under CGL perturbations, providing criteria for stability and instability.

Contribution

It demonstrates spectral stability of NLS periodic solutions and establishes an instability criterion for their persistence in CGL.

Findings

Periodic NLS solutions are spectrally stable under periodic perturbations.

An instability criterion for solutions persisting in CGL is derived.

The Fredholm alternative is used to analyze stability and instability conditions.

Abstract

We consider periodic traveling wave solutions to the focusing nonlinear Schrodinger equation (NLS) that have been shown to persist when the NLS is perturbed to the complex Ginzburg-Landau equation (CGL). In particular, we show that these periodic traveling waves are spectrally stable solutions of NLS with respect to periodic perturbations. Furthermore, we use an argument based on the Fredholm alternative to find an instability criterion for the persisting solutions to CGL.