Abstract
In this thesis criticality theory and conformal mapping schemes are established for single and two layer water-wave systems. Models describing these systems are introduced and their basic properties, namely dispersion relation, shallow water and Boussinesq approximations, and multi-symplectic formulations are derived. The theory of uniform flows and criticality is introduced using the shallow water equations associated with each of the models. Uniform flow solutions are then represented as surfaces with the criticality regions identified. These results are then recovered using the multi-symplectic formulations of the Boussinesq and full equations of each of the water-wave Systems. In this framework uniform flows are characterised as relative equilibria. The concept of criticality is then extended to non-trivial steady states by considering periodic waves coupled to uniform flows. Criticality conditions are defined for these flows which we call secondary criticality. A conformal mapping framework is established for a single layer flow and a two layer flow with a rigid lid. Firstly, a general parameterisation of the free surface and interface is considered. The governing equations are restated in terms of the new coordinates and are shown to be Hamiltonian. A mapping is then introduced for the fluid such that the parametric forms of the governing equations stated initially are recovered. Integral boundary relations are obtained relating the real and imaginary parts of periodic analytic functions. The conformal mapping schemes are then used to compute travelling waves. This is achieved by composing two numerical schemes which distinguish between the cases of finite and infinite layer, depths.