2016/12/28 by Walter Craig, Craig, Walter
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #35L72 #35Q31 #76B15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.1612.08971
openalex publication_date 2016/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.