Existence of gravity-capillary Crapper waves with concentrated vorticity

TL;DR

This paper proves the existence of gravity-capillary Crapper waves with concentrated vorticity, including point vortices and vortex patches, demonstrating that such waves can be overhanging under small gravity and vorticity conditions.

Contribution

It establishes the existence of these specialized waves with concentrated vorticity, a novel result in the study of water wave theory.

Findings

Existence of gravity-capillary Crapper waves with concentrated vorticity.

Waves can be overhanging with small gravity and vorticity.

Inclusion of point vortex and vortex patch models.

Abstract

The aim of this paper is to prove the existence of gravity-capillary Crapper waves with the presence of vorticity. In particular, we consider a concentrated vorticity: point vortex and vortex patch. We show that for small gravity and small vorticity it is possible to demonstrate that the waves are overhanging.