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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:
35L60Nonlinear first-order hyperbolic equations
35Q55NLS-like (nonlinear Schrödinger) equations
37G05Normal forms