About plane periodic waves of the nonlinear Schr\"odinger equations
Abstract
The present contribution contains a quite extensive theory for the stability analysis of plane periodic waves of 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. 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.
Keywords
Cite
@article{arxiv.2105.08341,
title = {About plane periodic waves of the nonlinear Schr\"odinger equations},
author = {Corentin Audiard and L Rodrigues},
journal= {arXiv preprint arXiv:2105.08341},
year = {2021}
}