Large-Amplitude Steady Solitary Water Waves with General Vorticity

TL;DR

This paper investigates large-amplitude steady solitary water waves with general vorticity, using conformal mappings and bifurcation theory to establish existence results for both small and large waves.

Contribution

It introduces a novel reformulation of the water wave problem allowing general vorticity and overhanging profiles, and proves existence of large-amplitude waves via global bifurcation.

Findings

Existence of small-amplitude solitary waves proven.

Existence of large-amplitude solitary waves established.

Reformulation enables handling of general vorticity and overhanging profiles.

Abstract

In this paper, we study two-dimensional steady solitary gravity waves propagating along the surface of a fluid of finite depth. In particular, we can deal with general vorticity distributions and overhanging wave profiles. By conformal mappings, we reformulate the problem into an overdetermined elliptic system coupled with an elliptic boundary value problem in a fixed strip domain. To avoid imposing extra constraints on vorticity function, we further reformulate the problem into the form of an abstract operator. Based on the formulations, the existence of small-amplitude solitary waves is proved by the center manifold reduction method, while the large-amplitude waves are obtained based on the analytic global bifurcation theorem.