Weakly Nonlinear Long Wave Models in Stratified Fluids

While there is a large literature concerning one - and two-dimensional weakly nonlinear long waves in fluid systems, there has been relatively little work which relates the many models describing such systems. It is the purpose of this thesis to construct a general model for such systems with this broader view in mind. This construction will resolve several problems in the theory of long wave propagation. In addition, it will provide new evolution equations which are of interest in their own right. In Chapter 1 we give a brief history of the study of long wave propagation in fluids as it relates to our study. The second chapter studies the dispersion relations of the linearized models. This is useful both from a pedagogical point of view and to help the regions of validity of the models. In Chapter 3, the core of the thesis, a "master" -Boussinesq system is derived which contains the many known evolution equations for weakly nonlinear long waves as special cases. Several new equations are also obtained. Chapter 4 contains a general derivation of single solitary wave solutions to many of the equations obtained in the previous chapter. One of the problems in solitary wave theory is the nature of the shallow water limit of equations governing waves in fluid of intermediate depth. The solutions obtained in Chapter 4 show precisely how the solitary waves scale in this limit. In Chapter 5 we use Hirota's method to analyze the two-dimensional Boussinesq equation. This new and interesting equation shares some of the special properties of the Kadomstev-Petviasvili equation and, in a certain sense, is more relevant physically. Finally, Chapter 6 contains a discussion of remaining open questions and issues raised by this work.