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Justification of modulation equations for hyperbolic systems via normal forms. (English) Zbl 0890.35082

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:

35L60 First-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37G05 Normal forms for dynamical systems
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