Gravity capillary standing water waves

TL;DR

This paper constructs small amplitude, standing water wave solutions in a 2D gravity-capillary system, demonstrating existence for various surface tensions and frequencies using advanced analytical techniques.

Contribution

It provides the first existence proof of standing wave solutions for the nonlinear gravity-capillary water waves equations with Sobolev regularity.

Findings

Existence of small amplitude standing waves for almost all surface tensions.

Solutions are periodic in time and space, not traveling.

Application of Nash-Moser scheme and microlocal analysis techniques.

Abstract

The paper deals with the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the question of the nonlinear interaction of a plane wave with its reflection off a vertical wall. The main result is the construction of small amplitude, standing (namely periodic in time and space, and not travelling) solutions of Sobolev regularity, for almost all values of the surface tension coefficient, and for a large set of time-frequencies. This is an existence result for a quasi-linear, Hamiltonian, reversible system of two autonomous pseudo-PDEs with small divisors. The proof is a combination of different techniques, such as a Nash-Moser scheme, microlocal analysis, and bifurcation analysis.