Hamiltonian Methods in Hydrodynamics

In 1984, Benjamin proposed a Hamiltonian structure for the Boussinesq model in the two dimensional context. The model describes the motion of an incompressible inviscid fluid with non uniform density, and is used to study internal wave phenomena. The Hamiltonian structure he derived has a peculiarity: in the presence of rigid wall delimiting the fluid domain, it breaks down if the density is not constant along them. This singular behaviour has been investigated later on by Camassa et al. (2014). In that study it is pointed out that the topology of initial conditions can affect the set of conserved quantities of the system. Specifically, the authors proved that for initial conditions with non uniform density along the boundary of the fluid, i.e. with disconnected isopycnals, the system retains only some of the conserved quantities it would have in the opposite case. This behavior, called "topological selection of conserved quantities", is studied in detail in the special case of a stratified fluid composed of two layers with different constant densities. The aim of the present work is to investigate the dynamical transition between configurations having different sets of conserved quantities. From a physical point of view, a satisfactory model has to allow topological changes of the flow, such as disconnection or re-connection of isopycnals (for example, think about a bubble of air that emerges from the water surface). However, the phenomenon of topological selection of the conserved quantities would seem to suggest that such transitions are not allowed by the models. We study the question in the context of one dimensional shallow water equations with two fluid layers. Specifically we investigate the dynamical interaction between the surfaces delimiting the two fluids. This aim is pursued considering a special class of solutions of the model, which provides polynomial field variables. In this setting is proved that the above surfaces can not come in contact, nor can detach, during the time evolution of the system. The present work is structured as follows: in the first chapter are summarized some basics about symmetries, conservation laws and Hamiltonian structures of differential equations, to be exploited in the subsequent chapters. The exposition is based on the books of Olver (1993) and Krasil'shchik and Vinogradov (1999). In chapter two are resumed the works of Benjamin (1984) and Camassa et al. (2014). Finally, in chapter three is addressed the study of the shallow water model.