On the Hamiltonian for water waves

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. 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. In this article we generalize the Hamiltonian formulation of water waves by Zakharov to a general coordinatization of the dynamical free surface, which allows it to apply to situations that include overturning wave profiles. This answers a question posed to the author by T.~Nishida during the RIMS Symposium on Mathematical Analysis in Fluid and Gas Dynamics that took place during July 6 - 8 2016.

arXiv comment10 pages

Cited by in corpus (4)

• A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension
• A Hamiltonian structure of the Isobe-Kakinuma model for water waves
• A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid
• Water wave problem with inclined walls