Hamilton System Theory, Including Wave Study

The theory of internal waves in stratified layers of fluid is important both for its interest to ocean engineering, and as a source of numerous interesting mathematical model equations which exhibit nonlinearity and dispersion. In this paper, using an expression for Dirichlet integral in terms of the Dirichlet-Neumann operator, we derive a Hamiltonian formulation of the fluid in terms of Zakharov's Hamiltonian. From the formulation we carry out a systematic analysis of the principal long wave scaling regimes. Our considerations include the Boussinesq and the KdV regimes. Our formulation of the fluid is shown to be very effective for perturbation calculations, and as well it holds promise as a basis for numerical simulations.The main conclusions of this dissertation are described as follows.1. We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation, using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators. In addition, for the convenience and simplicity of formulas' deducing, we suppose the upper layer is rigid lid. Certainly, using the Taylor expansions of the Dirichlet-Neumann operators for the upper fluid domains in the appendix of chapter 2, our methods can be extended to the free surface case.2. We study the long-wave asymptotic regime for internal waves in three bodies of immiscible fluid with rigid lid upper boundary conditions. We derive a Hamiltonian formulation for two-dimensional nonlinear long waves. From the Formulation, we obtain the linearized free interfaces equation. Then we get the corresponding dispersion relation and know that there are two different modes of waves motion, namely two interfaces displacements. In addition, using the Hamiltonian perturbation theory, we get the fully coupled effective Boussinesq equations of two interfaces. According to the difference between the interfaces, we derive respectively the KdV equation of a interface and the coupled KdV equation of the two interface in regime emphasizing one-way propagation. The computations for the results are performed in the framework of an asymptotic analysis of multiple scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.