Abstract

The analysis of traveling waves for the water waves problem has been tackled for more than a century by many mathematicians. From Stokes’s pioneering work in 1847 to the first rigorous result by Levi Civita in 1925 and until improvements in recent years, all research so far has mainly concerned the existence of periodic traveling waves, which are steady with respect to a moving frame.

In this talk I present a recent result about the existence of small amplitude, quasi-periodic in time traveling waves solutions for space periodic gravity-capillary water with constant vorticity in two dimensions. As far as we know, this is the first result of quasi-periodic in time traveling water waves and it is much more difficult than the periodic case due to the presence of small divisors.

In the first part I will introduce the water waves system starting from the Euler equations and I will explain the main statement, while in the second part I will give some glimpses of the proof, together with some analytical techniques.

This is a joint work with Massimiliano Berti and Alberto Maspero.