Hamiltonian models for the propagation of long gravity waves, higher-order KdV-type equations and integrability

TL;DR

This paper develops Hamiltonian models for long gravity wave propagation, deriving higher-order KdV-type equations that incorporate nonlinear and dispersive effects, and explores their integrability properties.

Contribution

It introduces a Hamiltonian framework for long gravity waves and derives higher-order KdV-type equations with explicit connections to integrable PDEs.

Findings

Derivation of Hamiltonian form for surface wave equations

Development of higher-order KdV-type equations with nonlinear and dispersive terms

Establishment of explicit transformations linking HKdV to integrable PDEs

Abstract

A single incompressible, inviscid, irrotational fluid medium bounded above by a free surface is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to a KdV approximation with higher order nonlinearities and dispersion (higher-order KdV-type equation, or HKdV). The HKdV is related to the known integrable PDEs with an explicit nonlinear and nonlocal transformation.