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.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Received by the editors February 26, 2009; accepted for publication (in revised form) December 21, 2009; published electronically March 12, 2010. This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01) and by award KUK-I1-007-43, provided by King Abdullah University of Science and Technology (KAUST).Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, England. This author's research was supported by a Royal Society University Research Fellowship (email@example.com).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.