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

• 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.
Mathematics Subject Classification: Primary: 35C07, 35B32; Secondary: 35A16, 35R35, 76B45, 76B70.

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