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1 April 2001 Long-wave short-wave resonance for nonlinear geometric optics
Thierry Colin, David Lannes
Duke Math. J. 107(2): 351-419 (1 April 2001). DOI: 10.1215/S0012-7094-01-10725-4

Abstract

The aim of this paper is to study oscillatory solutions of nonlinear hyperbolic systems in the framework developed during the last decade by J.-L. Joly, G. Métivier, and J. Rauch. Here we focus mainly on rectification effects, that is, the interaction of oscillations with a mean field created by the nonlinearity. A real interaction can occur only under some geometric conditions described in [JMR1] and [L1] that are generally not satisfied by the physical models except in the 1-dimensional case. We introduce here a new type of ansatz that allows us to obtain rectification effects under weaker assumptions. We obtain a new class of profile equations and construct solutions for a subclass. Finally, the stability of the asymptotic expansion is proved in the context of Maxwell-Bloch-type systems.

Citation

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Thierry Colin. David Lannes. "Long-wave short-wave resonance for nonlinear geometric optics." Duke Math. J. 107 (2) 351 - 419, 1 April 2001. https://doi.org/10.1215/S0012-7094-01-10725-4

Information

Published: 1 April 2001
First available in Project Euclid: 5 August 2004

zbMATH: 1034.35133
MathSciNet: MR1823051
Digital Object Identifier: 10.1215/S0012-7094-01-10725-4

Subjects:
Primary: 35Q60
Secondary: 35B34 , 35B40 , 35C20 , 35L60

Rights: Copyright © 2001 Duke University Press

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Vol.107 • No. 2 • 1 April 2001
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