Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations

Rémi Carles, Eric Dumas, Christof Sparber

Research output: Contribution to journalArticlepeer-review

22 Scopus citations

Abstract

We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrödinger 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 of the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrödinger equation on the torus in negative order Sobolev spaces. © 2010 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)489-518
Number of pages30
JournalSIAM Journal on Mathematical Analysis
Volume42
Issue number1
DOIs
StatePublished - Jan 2010
Externally publishedYes

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