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

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.

arXiv commentTo appear on Memoirs of the American Mathematical Society

References in corpus (4)

• On the Water Waves Equations with Surface Tension
• Time quasi-periodic gravity water waves in finite depth
• KAM for autonomous quasi-linear perturbations of mKdV
• Quasi-periodic solutions for fully nonlinear forced reversible Schroedinger equations

Cited by in corpus (5)

• Full description of Benjamin-Feir instability of Stokes waves in deep water
• Spatially quasi-periodic water waves of infinite depth
• Quasi-periodic traveling gravity-capillary waves
• About linearization of infinite-dimensional Hamiltonian systems
• Bifurcation of gravity-capillary Stokes waves with constant vorticity