Justification of the 2D NLS equation for a fourth order nonlinear wave equation – quadratic resonances do not matter much in case of analytic initial conditions

Abstract

The 2D Nonlinear Schrödinger (NLS) equation can be derived by multiple scaling analysis in order to describe slow temporal and spatial modulations of an oscillating wave packet in general dispersive wave systems. It is the purpose of this paper to establish estimates for the difference between the so-obtained approximation and true solutions of the original system in case that the original system possesses nontrivial quadratic resonances. We explain for a fourth order nonlinear wave equation that quadratic resonances do not matter much in case of analytic initial conditions for the NLS equation in the sense that independently of whether the resonance is stable or unstable an approximation property can be established on the natural time scale of the NLS equation if the set of resonant wave numbers is bounded away from the integer multiples of the basic wave number of the underlying wave packet. Finally, we explain that in case of an approximation by a so-called four wave interaction (FWI) system the same approach works and an even stronger result can be established.