A Hamiltonian approach to the CH-KP equation

Abstract

A model governing quasi-unidirectional wave propagation at the surface of a shallow layer of water is derived from Euler equations by long wave asymptotics together with Hamiltonian reduction techniques. The balance between nonlinearity, dispersion and weak dependence on the second planar coordinate is examined and results in the so-called CH-KP equation, a mildly nonlinear version of the well known Kadomtsev-Petviashvili equation. The Hamiltonian structure of the model is obtained by Dirac reduction from the intermediate step of a fully-nonlinear, long-wave system for stratified fluids. A particular scaling is considered that reduces such system to the one-dimensional CH equation in a limiting case. Examples of weak peakon-type solutions are provided, chosen to illustrate the collision behavior supported by the limiting equation.