Rational Solutions of Nonlinear Partial Differental Equations

The work in this thesis considers rational solutions of nonlinear partial differential equations formed from polynomials. The main work will be on the Boussinesq equation and the Kadomtsev-Petviashvili-I (KP-I) equation, the nonlinear Schroedinger equation will also be included for completeness.

Rational solutions of the Boussinesq equation model rogue wave behaviour. These solutions are shown to be highly structured which, it is hypothesised, is due to the inherent structure and form of integrable differential equations. Rogue wave solutions have been observed in equations such as the nonlinear Schr\"odinger equation, KP equation and the Boussinesq equation, to name but a few. By examining the form of these solutions and considering the behaviour of the roots, the aim is to establish the behaviour of this family of solutions. All solutions are bounded and real.

Additionally, since a generating function for the KP equation solutions already exists, a characterisation of the solutions will be made along with an attempt at understanding the current generating function in order to improve its adaptability.

Links between solutions of the three equations will be shown as well as a function that can solve all three equations subject to certain criteria on the parameters.