Peaked Stokes waves as solutions of Babenko's equation

TL;DR

This paper demonstrates that Babenko's equation can describe peaked Stokes waves in deep water, revealing singularity properties and ruling out certain corner point singularities in holomorphic coordinates.

Contribution

It extends the application of Babenko's equation to peaked waves and analyzes their singularity structure in the deep-water limit.

Findings

Peaked Stokes waves can be characterized using Babenko's equation.

The analysis rules out corner point singularities in holomorphic coordinates.

Properties of peaked waves are recoverable from Babenko's equation in the deep-water limit.

Abstract

Babenko's equation describes traveling water waves in holomorphic coordinates. It has been used in the past to obtain properties of Stokes waves with smooth profiles analytically and numerically. We show in the deep-water limit that properties of Stokes waves with peaked profiles can also be recovered from the same Babenko's equation. In order to develop the local analysis of singularities, we rewrite Babenko's equation as a fixed-point problem near the maximal elevation level. As a by-product, our results rule out a corner point singularity in the holomorphic coordinates, which has been obtained in a local version of Babenko's equation.