An existence theory for small-amplitude doubly periodic water waves with vorticity

TL;DR

This paper establishes the existence of three-dimensional steady gravity-capillary water waves with vorticity, periodic in two dimensions, using multi-parameter bifurcation theory, expanding understanding of complex wave behaviors with vorticity.

Contribution

It introduces an existence theory for small-amplitude doubly periodic water waves with vorticity, specifically with Beltrami velocity fields, using bifurcation methods.

Findings

Existence of three-dimensional steady gravity-capillary waves with vorticity.

Waves are periodic with respect to a two-dimensional lattice.

The velocity field is a Beltrami field, with vorticity collinear to velocity.

Abstract

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is collinear to the velocity. The existence theory is based on multi-parameter bifurcation theory.