A Study of Rotational Water Waves using Bifurcation Theory

Abstract

This thesis is concerned with the water wave problem. Using local bifurcation we establish small-amplitude steady and periodic solutions of the Euler equations with vorticity. Our approach is based on that of Ehrnström, Escher and Wahlén \cite{EEW11}, the main difference being that we use new bifurcation parameters. The bifurcation is done both from a one-dimensional and a two-dimensional kernel, the latter bifurcation giving rise to waves having more than one crest in each minimal period. We also give a novel and rudimentary proof of a key lemma establishing the Fredholm property of the elliptic operator associated with the water wave problem. Furthermore, we investigate derivatives of the bifurcation curve, and present a new result for the corresponding linear problem.