Interaction of modulated gravity water waves of finite depth

Abstract

We consider the capillary-gravity water-waves problem of finite depth with a flat bottom of one or two horizontal dimensions. We derive the modulation equations of leading and next-to-leading order in the hyperbolic scaling for three weakly amplitude-modulated plane-wave solutions of the linearized problem in the absence of quadratic and cubic resonances. We fully justify the derived system of macroscopic equations in the case of pure gravity waves, i.e. in the case of zero surface tension, employing the stability of the water-waves problem on the time-scale $O(1/\ep)$ obtained by Alvarez-Samaniego and Lannes.