Criticality theory and conformal mapping techniques for single and two layer water-wave systems
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