ON THE HAMILTONIAN FOR WATER WAVES (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

Many equations that arise in a physical context can be posed in the form of a Hamiltonian system, meaning that there is a symplectic structure on an appropriate phase space, and a Hamiltonian functional with respect to which time evolution of their solutions can be expressed in terms of a Hamiltonian vector field. A normal forms transformation for a Hamiltonian dynamical system given by such a vector field is a change of variables in a neighborhood of a stationary point in phase space that eliminates inessential terms, retaining only essential nonlinearities while preserving the Hamiltonian structure of the system. It is known from the work of VE Zakharov that the equations for water waves can be posed as a Hamiltonian dynamical system, and that the equilibrium solution is an elliptic stationary point. This article discusses two aspects of the water wave equations in this context. Firstly, we generalize the Hamiltonian formulation of Zakharov to include overturning wave profiles, answering a question posed to the author by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms transformations for the water waves of equations, in the setting of spatially periodic solutions. Our results describe the function space mapping properties of the normal forms transformations, with and without inclusion of the effects of surface tension, and the dynamical implications of the normal forms. This latter is joint work with Catherine Sulem (University of Toronto).