Local bifurcation of steady almost periodic water waves with constant vorticity

2018/05/23 by Luo, Wei, Yin, Zhaoyang
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.08987

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 \citeConstantin2011v. Moreover, our result implies a non-uniqueness result at the same bifurcation point.

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