Quasi-periodic traveling waves on an infinitely deep fluid under gravity

TL;DR

This paper proves the existence and linear stability of small amplitude, quasi-periodic traveling water waves in an infinite-depth setting, using advanced mathematical techniques to handle resonances and nonlinearities.

Contribution

It is the first to establish the existence of quasi-periodic water waves bifurcating from a resonant elliptic fixed point, employing a novel normal form approach and Hamiltonian structure analysis.

Findings

Existence of quasi-periodic traveling water waves in infinite depth.

Linear stability of these quasi-periodic solutions.

Development of a new normal form method for resonant interactions.

Abstract

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a \emph{completely resonant} elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with the combined problems of \emph{small divisors} and the \emph{fully-nonlinear} nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined \emph{nonlinear} bifurcation analysis involving several non-trivial resonant wave interactions, as the well-known "Benjamin-Feir resonances". We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.