Interaction of modulated gravity water waves of finite depth

TL;DR

This paper derives and justifies modulation equations for weakly amplitude-modulated water waves of finite depth, focusing on pure gravity waves without surface tension, using stability analysis.

Contribution

It provides a rigorous derivation and justification of modulation equations for gravity water waves of finite depth in the hyperbolic scaling regime.

Findings

Derived modulation equations for water waves of finite depth.

Justified the equations in the pure gravity wave case.

Established stability results for the water-waves problem.

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.