IUMJ

Title: Justification of the nonlinear Schroedinger equation for a resonant Boussinesq model

Authors: Wolf-Patrick Duell and Guido Schneider

Issue: Volume 55 (2006), Issue 6, 1813-1834

Abstract:

The nonlinear Schroedinger equation formally describes slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. It is the purpose of this paper to prove estimates between the formal approximation, obtained via the nonlinear Schroedinger equation, and true solutions of the original system. The method developed in an earlier paper in case of non-trivial quadratic resonances is improved to cover also the additional problem of a trivial resonance at the wavenumber $k = 0$ as it occurs for the water wave problem. For a Boussinesq equation, a formal and phenomenological model for surface water waves subject to gravity and surface tension, we establish the approximation property in case the formal NLS approximation is stable in the system for the three wave interaction associated to the resonance. Although we restrict ourselves to a Boussinesq equation, we believe that the result is also true for the full water wave problem.