Hamiltonian formulation of the extended Green–Naghdi equations

• We derive the extended Green–Naghdi (GN) equations which incorporate the higher-order dispersion.
• We show that the extended GN equations have the same Hamiltonian structure as that of the GN equation.
• We prove that Zakharov’s Hamiltonian equations of motion are equivalent to the extended GN equations.

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.