About plane periodic waves of the nonlinear Schr\"odinger equations

TL;DR

This paper develops a comprehensive stability theory for plane periodic waves in nonlinear Schrödinger equations, covering one-dimensional and multi-dimensional cases, and demonstrates their spectral instability in various regimes.

Contribution

It extends stability analysis of plane periodic waves to multi-dimensional Schrödinger equations, including spectral and nonlinear stability results, with new modulation asymptotics and instability proofs.

Findings

Spectral instability in small-amplitude regimes

Spectral instability in large-period regimes

Unified stability theory for one- and multi-dimensional cases

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.