Hamiltonian long-wave approximations to the water-wave problem

This paper presents a Hamiltonian formulation of the water-wave problem in which the non-local Dirichlet-Neumann operator appears explicitly in the Hamiltonian. The principal long-wave approximations for water waves are derived by the systematic approximation of the Dirichlet-Neumann operator by a sequence of differential operators obtained from a convergent Taylor expansion of the Dirichlet-Neumann operator. A simple and satisfactory method of obtaining the classical two-dimensional approximations such as the shallow-water, Boussinesq and KdV equations emerges from the process. A straightforward transformation theory describes the relationship between the classical symplectic structure appearing in the water-wave problem and the various nonclassical symplectic structures that arise in long-wave approximations. The discussion extends to include three-dimensional approximations, including the KP equation.