NDLI: Multiphase Weakly Nonlinear Geometric Optics for Schrdinger Equations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Carles, Rmi Dumas, Eric Sparber, Christof
Copyright Year	2010
Abstract	We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrdinger 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 Schrdinger equation on the torus in negative order Sobolev spaces.
Starting Page	489
Ending Page	518
Page Count	30
File Format	PDF
ISSN	00361410
DOI	10.1137/090750871
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	1
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-03-12
Access Restriction	Subscribed
Subject Keyword	instability NLS-like equations resonances nonlinear Schrdinger equation high-frequency limit Asymptotic behavior of solutions
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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