Local bifurcation of steady almost periodic water waves with constant vorticity

TL;DR

This paper studies bifurcation phenomena in steady almost periodic water waves with constant vorticity, using conformal mappings and pseudodifferential equations to establish existence and non-uniqueness results.

Contribution

It generalizes previous bifurcation results for water waves to almost periodic cases and demonstrates non-uniqueness at bifurcation points.

Findings

Existence of bifurcating solutions for almost periodic water waves.

Extension of prior results to more general wave profiles.

Identification of non-uniqueness at bifurcation points.

Abstract

In this paper we mainly investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem with some unknown functions in the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for a almost periodic function of one variable. The bifurcation theory ensures us to obtain a existence result. Our existence result generalizes and covers the recent result in \cite{Constantin2011v}. Moreover, our result implies a non-uniqueness result at the same bifurcation point.