Nonlinear and spectral stability of periodic traveling wave solutions for a nonlinear Schrödinger system

Abstract

This paper is concerned with nonlinear and spectral stability of periodic traveling wave solutions for a nonlinear Schrödinger type system arising in nonlinear optics. We prove the existence of two smooth curves of periodic solutions depending on the cnoidal type functions. In the framework established by Grillakis, Shatah and Strauss we prove a stability result under perturbations having the same minimal wavelength and zero mean over their fundamental period. By using the so-called Bloch wave decomposition theory we show spectral stability for a general class of periodic solutions.