Multiphase weakly nonlinear geometric optics for Schrodinger equations

Details Multiphase weakly nonlinear geometric optics for Schrodinger equations Multiphase weakly nonlinear geometric optics for Schrodinger equations

2009-02-14

arxiv.org/abs/0902.2468v2

SIAM Journal on Mathematical Analysis / SIAM Journal of Mathematical Analysis 42, 1 (2010) 489-518

Mathematics

Analysis of PDEs

29 pages

Scientific paper

10.1137/090750871

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.

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