Multiphase weakly nonlinear geometric optics for Schrodinger equations

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.

29 pages

References in corpus (2)

• Theory of four-wave mixing of matter waves from a Bose-Einstein condensate
• Semiclassical asymptotics for weakly nonlinear Bloch waves

Cited by in corpus (6)

• Energy cascades for NLS on the torus
• On the one-dimensional cubic nonlinear Schrodinger equation below L^2
• Nonlinear coherent states and Ehrenfest time for Schrodinger equation
• Norm inflation for nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces of negative regularity
• On nonlinear effects in multiphase WKB analysis for the nonlinear Schrodinger equation
• Interaction of coherent states for Hartree equations