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

Samuel Walsh

doi:10.3934/dcds.2014.34.3241

Article Contents

2014, Volume 34, Issue 8: 3241-3285. Doi: 10.3934/dcds.2014.34.3241

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

Samuel Walsh^1,

1.
University of Missouri, Columbia, MO 65201

Received: June 30, 2013

Revised: August 31, 2013

Published: July 2014

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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.

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

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