Large-amplitude steady solitary water waves with constant vorticity

TL;DR

This paper constructs large-amplitude steady solitary water waves with constant vorticity, allowing for complex features like stagnation points and overhanging profiles, using advanced bifurcation theory and a novel elliptic reformulation.

Contribution

It introduces a new analytical approach to prove existence of large-amplitude solitary waves with constant vorticity, including waves with internal stagnation points and overhanging profiles.

Findings

Existence of continuous solution curves with unbounded wave speed

Solitary waves of elevation with constant vorticity are supercritical

Development of singularities in the conformal map at maximum amplitude

Abstract

This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave.