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