S. Locke and D.E. Pelinovsky
Peaked Stokes waves as solutions of Babenko's equation
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.
Keywords:
Babenko's equation; Stokes waves; existence of peaked traveling waves; fixed-point methods.