Aspects of the mathematical theory of water waves

Abstract

In this thesis we study various aspects of the mathematical theory of water waves. In Chapter 2 some qualitative results for two recently-derived nonlinear models for shallow water waves are presented. In the first part of the chapter we examine the solutions of a family of nonlinear differential equations which initially have compact support, and investigate whether they retain this property over a nontrivial time interval. In the second section of Chapter 2 we particularise to the Degasperis-Procesi equation and conduct a more detailed analysis of the propagation speed of solutions for a larger class of initial data, achieving results on the persistence properties of solutions and the unique continuation of solutions.