Time quasi-periodic traveling gravity water waves in infinite depth

Abstract

We present the recent result [9] 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.