On the Spectral Stability of Solitary Wave Solutions of the Vector Nonlinear Schrödinger Equation

We consider a system of coupled cubic nonlinear Schrödinger (NLS) equations i ∂ψj ∂t = − ∂ψj ∂x2 + ψj n ∑ k=1 αjk|ψk| j = 1, 2, . . . , n, where the interaction coefficients αjk are real. The spectral stability of solitary wave solutions (both bright and dark) is examined both analytically and numerically. Our results build on preceding work by Nguyen et al. and others. Specifically, we present closed-form solitary wave solutions with trivial and non-trivial phase profiles. Their spectral stability is examined analytically by determining the locus of their essential spectrum. Their full stability spectrum is computed numerically using a large-period limit of Hill’s method. We find that all nontrivial-phase solutions are unstable while some trivial-phase solutions are spectrally stable. To our knowledge, this paper presents the first investigation of the stability of the solitary waves of the coupled cubic NLS equation without the restriction that all components ψj are proportional to sech.