Bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations

Abstract

We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger (CNLS) equations on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a fundamental solitary wave. We establish criteria under which the fundamental solitary wave undergoes a pitchfork bifurcation, and utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral and/or orbital stability of the bifurcated solitary waves as well as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to a cubic nonlinearity case and give numerical evidences for the theoretical results.