Time quasi-periodic gravity water waves in finite depth

TL;DR

This paper proves the existence and linear stability of small amplitude, time quasi-periodic standing water waves in finite depth, using KAM theory and pseudo-differential techniques to handle small divisors and quasi-linear challenges.

Contribution

It establishes the existence and stability of quasi-periodic water waves in finite depth for a full measure set of depths, employing novel reducibility and KAM methods.

Findings

Existence of Cantor families of quasi-periodic solutions

Linear stability of these solutions

Applicable to almost all depth parameters in a measure-theoretic sense

Abstract

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.