Multiphase weakly nonlinear geometric optics for Schrodinger equations

TL;DR

This paper develops a rigorous geometric optics framework for analyzing highly oscillatory solutions of nonlinear Schrödinger equations, revealing wave interaction mechanisms and extending instability results in negative Sobolev spaces.

Contribution

It introduces a multiphase weakly nonlinear geometric optics approach for Schrödinger equations, providing detailed analysis of wave mixing and resonance structures.

Findings

Rigorous justification of nonlinear wave interactions.

Analysis of resonance structures in cubic nonlinearities.

Extension of instability results in negative Sobolev spaces.

Abstract

We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrodinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slowly modulated plane waves, and use the framework of Wiener algebras. A detailed analysis of the corresponding nonlinear wave mixing phenomena is given, including a geometric interpretation on the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrodinger equation on the torus in negative order Sobolev spaces.