An asymptotic behaviour near the crest of waves of extreme form on water of finite depth

TL;DR

This paper analyzes the local asymptotic behavior of extreme water waves with vorticity near stagnation points, revealing how vorticity influences wave profile regularity and shape, especially near the crest.

Contribution

It provides new higher-order asymptotics and regularity results for extreme water waves with vorticity, applicable to various wave types and depths.

Findings

Extreme waves with negative vorticity are concave near the crest.

Wave behavior and regularity depend significantly on vorticity distribution.

Results are broadly applicable to different wave and flow configurations.

Abstract

We prove local higher-order asymptotics for extreme water waves with vorticity near stagnation points. We obtain that the behaviour of solutions and their regularity depend substantially on the vorticity. In particular, we show that extreme waves with a negative vorticity distribution have concave profiles near the crest. Our approach is based on new regularity results and asymptotic analysis of the corresponding nonlinear problem in a half-strip. Our main result is local and therefore is valid for a broad range of problems, such as for waves with a piecewise constant vorticity, stratified waves, flows with counter-currents or waves on infinite depth.