Quasi-periodic standing wave solutions of gravity-capillary water waves

TL;DR

This paper proves the existence and linear stability of small amplitude quasi-periodic standing wave solutions in a 2D gravity-capillary water wave model, valid for most surface tension values.

Contribution

It establishes the existence and stability of quasi-periodic standing waves for 2D water waves with surface tension, covering a full measure set of tension values.

Findings

Existence of quasi-periodic standing wave solutions.

Linear stability of these solutions.

Validity for a full measure set of surface tension values.

Abstract

We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.