| In this dissertation, algebraic geometric methods are used to study new soliton dynamics such as soliton fusion, fission, and change of form. These methods are also used to find piecewise solutions to the Camassa Holm shallow water equation from shallow water theory.; This work begins with a brief overview of the history and techniques of integrable partial differential equations. These are the nonlinear evolution equations which have an associated Lax pair. Next, the algebraic geometric approach is introduced. In particular, the method of recurrence chains is demonstrated to be a shortcut to finding solutions using this approach. The Two-Component KdV equation is studied, and all possible one and two soliton solutions are found. This equation is linked to the physical models of shallow water theory and nonlinear optics. It is demonstrated through analysis of analytical representations for the solutions that there exist solutions having soliton fission, fusion, and change of form.; Next, the Camassa Holm shallow water equation is introduced. This equation is seen to possess peaked solitons whose derivative switches sign at the peaks. These peaked solutions are called peakons and exact formulas for single peakon and two peakon interaction are found using methods analogous to the ones used in level set theory.; The approach from level sets theory is generalized to find differential equations whose solutions describe the paths of solitons in the ( x, t) plane for any nonlinear evolution equation. Using these equations, a criterion is found which characterizes precisely which equations from a large class of equations, have the phenomena of soliton fission, fusion, and change of form.; Finally, it is demonstrated that additional terms may be added to any equation from this large class, which allows the equations to possess solitons having soliton fission, fusion, and change of form. These additional terms all contain a small parameter e , which when set to zero, recovers the original equation. |
