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Justification of the nonlinear Schrödinger equation for a resonant Boussinesq model. (English) Zbl 1194.35362
Summary: The nonlinear Schrödinger 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 Schrödinger 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.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 76D03 Existence, uniqueness, and regularity theory 76B03 Existence, uniqueness, and regularity theory (fluid mechanics)