Justification of modulation equations for hyperbolic systems via normal forms

Abstract.

The justification problem for the Nonlinear Schrödinger equation as a modulation equation for almost spatial periodic wavetrains 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.