Spatial hamiltonian structure, energy flux and the water-wave problem

Abstract

Travelling waves and relative equilibrium states of infinite dimensional hamiltonian systems are considered with particular attention to the water-wave problem. Relative equilibria correspond to solutions that are stationary in a moving frame of reference. The governing equations for relative equilibria are recast as a hamiltonian evolution equation in space. The mass, momentum and energy flux densities play a fundamental role in the spatial hamiltonian structure: the energy flux generates the basic symplectic operator, the momentum flux is the spatial hamiltonian and the mass flux is a second independent (spatial) integral. A complete theory is given for the water-wave problem with surface tension. The hamiltonian structure is used to show the existence of dual variational principles for travelling waves and to give an analysis of a singularity in the dispersion relation for capillary-gravity waves from a spatial hamiltonian systems viewpoint.