Overhanging solitary water waves

TL;DR

This paper constructs novel overhanging solitary gravity water waves with a disk and strip shape, using a gluing method that combines explicit solutions, advancing understanding of wave forms with constant vorticity.

Contribution

It introduces the first construction of overhanging solitary water waves with specific geometric features and constant vorticity, using a novel gluing technique.

Findings

Existence of overhanging solitary water waves with disk and strip shape.

Construction method combining explicit solutions and gluing techniques.

Applicable for small positive gravitational constant g.

Abstract

We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant $g>0$ is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, H\'elein, and Pacard \cite{hauswirth-helein-pacard}. The method developed here is related to the construction of constant mean curvature surfaces through gluing.