Local Bifurcation of Steady Almost Periodic Water Waves with Constant Vorticity

In this paper we 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 for some unknown functions with the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable. The bifurcation theory ensures that we can obtain an existence result. Our existence result generalizes and covers the recent result in [15]. Moreover, our result implies a non-uniqueness result at the same bifurcation point.