Geometric optics and instability for NLS and Davey-Stewartson models

TL;DR

This paper investigates high-frequency wave interactions in multi-dimensional nonlinear Schrödinger and Davey-Stewartson models, revealing localization phenomena and strong instability results in negative Sobolev spaces.

Contribution

It introduces a new localization phenomenon for non-local perturbations and demonstrates norm inflation with infinite loss of regularity in negative Sobolev spaces.

Findings

Discovery of a new localization phenomenon for non-local perturbations

Proof of strong instability and norm inflation in negative Sobolev spaces

Constructive approach to demonstrate infinite loss of regularity

Abstract

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrodinger equations with gauge invariant power-law nonlinearities and non-local perturbations. The model includes the Davey--Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variant. Our analysis reveals a new localization phenomenon for non-local perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.