Geometric Methods for some Nonlinear Wave Equations

Abstract: A number of results related to the geometric interpretation of some dispersive nonlinear wave equations are presented. It is first described how some well-known shallow water equations arise geometrically as Euler equations for the geodesic flow on the Virasoro group endowed with certain right-invariant metrics. For one of these equations?the Camassa-Holm equation?we demonstrate how geometric methods can be used to construct the infinite number of conservation laws and to establish stability of spatially periodic solutions under small perturbations of the initial data. The second half of the thesis is concerned with the geometric approach to the Hunter-Saxton equation?a model for the propagation of orientation waves in liquid crystal director fields. The equation is shown to be the Euler equation for the geodesic flow on an infinite-dimensional sphere, so that its solutions correspond simply to the big circles on the sphere. Using this geometric viewpoint several results are obtained. Most notably, by constructing a weak continuation of the geodesic flow, we show how to extend solutions of the periodic Hunter-Saxton equation beyond breaking time.