Multi-dimensional stability analysis of plane periodic waves of general Schr"odinger equations

About plane periodic waves of the nonlinear Schrödinger equations

We provide a quite extensive theory for the stabilityanalysis of plane periodic waves of general general schr{\"o}dinger equations.On one hand, we put the one-dimensional 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, and 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.