Time Quasi-Periodic Traveling Gravity Water Waves in Infinite Depth

TL;DR

This paper proves the existence of quasi-periodic traveling gravity water waves in infinite depth, using advanced mathematical techniques to handle resonances and nonlinearity in the water wave equations.

Contribution

It provides the first existence result of quasi-periodic water waves bifurcating from a resonant elliptic fixed point, employing a Nash-Moser scheme and normal form methods.

Findings

Existence of quasi-periodic solutions in infinite depth water waves

Application of Nash-Moser scheme to nonlinear water wave equations

Handling of small divisors and resonance issues in the proof

Abstract

We present the recent result [8] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result of quasi-periodic water waves solutions bifurcating from a 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 small divisors and the fully-nonlinear nature of the equations.