Pure gravity traveling quasi-periodic water waves with constant vorticity

TL;DR

This paper proves the existence of small amplitude, quasi-periodic traveling water wave solutions with constant vorticity, using a Nash-Moser scheme, for bidimensional fluids with a flat bottom and periodic free surface.

Contribution

It introduces a novel application of Nash-Moser iteration to construct quasi-periodic solutions in water waves with constant vorticity, expanding understanding of wave interactions.

Findings

Existence of small amplitude quasi-periodic solutions for water waves.

Solutions pass through each other, deforming slightly, and retain quasiperiodic structure.

Applicable for any fixed depth and gravity, with vorticity in a large measure set.

Abstract

We prove the existence of small amplitude time quasi-periodic solutions of the pure gravity water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space periodic free interface. Using a Nash-Moser implicit function iterative scheme we construct traveling nonlinear waves which pass through each other slightly deforming and retaining forever a quasiperiodic structure. These solutions exist for any fixed value of depth and gravity and restricting the vorticity parameter to a Borel set of asymptotically full Lebesgue measure.