We present the recent result  concerning the existence of quasi-periodic in time traveling waves for the 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.
Cite this article
Roberto Feola, Filippo Giuliani, Time quasi-periodic traveling gravity water waves in infinite depth. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 4, pp. 901–916DOI 10.4171/RLM/919