Hamiltonian formulation of the extended Green-Naghdi equations

Abstract

A novel method is developed for extending the Green-Naghdi (GN) shallow-water model equation to the general system which incorporates the arbitrary higher-order dispersive effects. As an illustrative example, we derive a model equation which is accurate to the fourth power of the shallowness parameter while preserving the full nonlinearity of the GN equation, and obtain its solitary wave solutions by means of a singular perturbation analysis. We show that the extended GN equations have the same Hamiltonian structure as that of the GN equation. We also demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

21 pages

References in corpus (1)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations

Cited by in corpus (3)

• Hamiltonian structure for two-dimensional extended Green-Naghdi equations
• Group classification of the two-dimensional Green-Naghdi equations with a time dependent bottom topography
• A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: Justification as a shallow water approximation