Validity of the resonant four-wave interaction system in a model for surface water waves on an infinite deep sea

Models for the resonant four-wave interaction play a big role in the description of gravity-driven surface water waves. The question however, if such models, really describe the evolution of surface water waves is a difficult task, according to the fact that the original water wave problem contains quadratic terms and a resonance at the wave vector k=0k=0. As a first step in answering the validity question positively we prove here that solutions of a model for surface water waves on an infinite deep sea really can be described by the associated systems for the resonant four wave interaction. We believe that the method developed in this paper applies to the problem of surface water waves on an infinite deep sea, too.