Integrable and non-integrable equations with peaked soliton solutions

This thesis explores a number of nonlinear PDEs that have peaked soliton solutions, to apply reductions to such PDEs and solve the resultant equations. Chapter 1 provides a brief history of peakon equations, where they come from and the different viewpoints of various authors. The rest of the chapter is then devoted to detailing the mathematical tools that will be used throughout the rest of the thesis. Chapter 2 concerns a coupling of two integrable peakon equations, namely the Popowicz system, which itself is not integrable. The 2-peakon dynamics are studied, and an explicit solution to the 2-peakon dynamics is given alongside some features of the interaction. In chapter 3 a reduction from two integrable peakon equations with quadratic nonlinearity to the third Painlev´e equation is given. B¨acklund transformations and solutions for the Painlev´e equations are expressed, and then used to find solutions of the original PDEs. A general peakon family, the b-family, is also explored, giving a more general result. Chapter 4 examines two peakon equations with cubic nonlinearity, and their reductions to Painlev´e equations. A link is shown between these cubic nonlinear peakon equations and the quadratic nonlinear equations in chapter 3. Chapter 5 has conclusions and outlook in the area.