Steady stratified periodic gravity waves with surface tension I: Local bifurcation

Abstract

In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of small-amplitude solutions. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from $\mathcal{T}$ at an eigenvalue of the linearized problem.