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We consider periodic traveling wave solutions to the focusing nonlinearSchrodinger equation (NLS) that have been shown to persist when the NLS isperturbed to the complex Ginzburg-Landau equation (CGL). In particular, we showthat these periodic traveling waves are spectrally stable solutions of NLS withrespect to periodic perturbations. Furthermore, we use an argument based on theFredholm alternative to find an instability criterion for the persistingsolutions to CGL.

Spectral stability of periodic NLS and CGL solutions