Steady periodic gravity waves with surface tension

TL;DR

This paper proves the existence of steady, periodic gravity-capillary waves with surface tension on stratified water, using bifurcation theory and global continuation methods.

Contribution

It introduces a novel global bifurcation framework for steady gravity-capillary waves with surface tension, combining degree theory and analytic continuation.

Findings

Existence of global continua of classical solutions

Construction of laminar flow solutions as bifurcation points

Application of Rabinowitz degree theory and Dancer's analytic continuation

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 global continua of classical solutions that are periodic and traveling. 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. Each solution curve is then continued globally by means of a degree theoretic theorem in the spirit of Rabinowitz. Finally, we complement the degree theoretic picture by proving an alternate global bifurcation theorem via the analytic continuation method of Dancer.