About plane periodic waves of the nonlinear Schrödinger equations

Abstract

The present contribution contains a quite extensive theory for the stability analysis of plane periodic waves of general Schrödinger equations. On one hand, we put the onedimensional theory, or in other words the stability theory for longitudinal perturbations, on a par with the one available for systems of Korteweg type, including results on co-periodic spectral instability, nonlinear co-periodic orbital stability, side-band spectral instability and linearized large-time dynamics in relation with modulation theory, and resolutions of all the involved assumptions in both the small-amplitude and large-period regimes. On the other hand, we provide extensions of the spectral part of the latter to the multi-dimensional context. Notably, we provide suitable multi-dimensional modulation formal asymptotics, validate those at the spectral level and use them to prove that waves are always spectrally unstable in both the small-amplitude and the large-period regimes.