A Hamiltonian formulation for strongly nonlinear internal waves.
Ricardo Barros,
(Instituto BCAM)
Jueves 16 de diciembre 2010
We will focus in this talk on the mathematical modeling of the internal wave phenomenon in fluids. We will give a variational formulation for a strongly nonlinear model for internal waves in a two-layer flow with a top free surface, and we will investigate its solitary-wave solutions. In the spirit of Whitham, the approximate theory consists on inserting approximations directly into the fluid Lagrangian, and it has two main advantages over the classical perturbation procedures. First, the approximations do not disturb the corresponding symmetry properties coming from the variational structure of the governing equations. Second, the approximation methods based on Hamilton’s principle suggest transformations to new dependent variables in which the approximate equations take its simplest mathematical form. The traveling-wave solutions are described by a Hamiltonian system with two degrees of freedom and we will investigate numerically the existence of true homoclinic orbits. Features such as the existence of two wave regimes, characterized by elevation or depression of the interface and the broadening of solutions are presented, and it is shown how these relate to a change of the global properties for the potential of the Hamiltonian system. Finally, other sets of parameters were considered for which multi-humped solutions exist, showing the richness and complexity of the Hamiltonian system considered here.