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Zbl 0890.35082
Schneider, Guido
Justification of modulation equations for hyperbolic systems via normal forms. (English)
[J] NoDEA, Nonlinear Differ. Equ. Appl. 5, No.1, 69-82 (1998). ISSN 1021-9722; ISSN 1420-9004/e

Summary: The justification problem for the nonlinear Schrödinger equation as a modulation equation for almost spatial periodic wave-trains of small amplitude is considered. We show exact estimates between solutions of the original system and their approximations which are obtained by the solutions of the nonlinear Schrödinger equation. By a normal form transform the a priori dangerous quadratic terms of the considered hyperbolic systems are eliminated. Then the transformed systems start with cubic terms. This allows to justify the nonlinear Schrödinger equation by a simple application of Gronwall's inequality. Moreover, the influence of resonances is estimated.

MSC 2000:: *35L60 First-order nonlinear hyperbolic equations
35Q55 NLS-like (nonlinear Schroedinger) equations
37G05 Normal forms

Keywords: almost spatial periodic wavetrains; resonances

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