Water waves with compactly supported vorticity: A functional-analytic approach to bifurcation theory and the mathematical theory of traveling water waves

Abstract

We study the mathematical theory of water waves. Local bifurcation theory is also discussed, including the Crandall-Rabinowitz theorem; an abstract theorem used to establish the presence of bifurcation points in the zero set of maps on Banach spaces. A functional-analytic approach is used to prove the existence of a family of localized traveling waves with one or more point vortices, by bifurcating from a trivial solution. This is done in the setting of the incompressible Euler equations with gravity and surface tension, on finite depth. Our result is an extension of a recent result by Shatah, Walsh and Zeng, where existence was shown for a single point vortex on infinite depth. The properties of the resulting waves are also examined: We find that the properties depend significantly on the position of the point vortices in the water column.