Large-amplitude steady solitary water waves with constant vorticity

Susanna V. Haziot, Miles H. Wheeler

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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. This is stronger than what one would expect from a straightforward generalization of existing results for periodic waves. 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.

Original languageEnglish
Article number27
Number of pages49
JournalArchive for Rational Mechanics and Analysis
Volume247
Issue number2
DOIs
Publication statusPublished - 21 Mar 2023

Bibliographical note

Funding Information:
The work of SVH is funded by the National Science Foundation through the award DMS-2102961.

Publisher Copyright:
© 2023, The Author(s).

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