  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.
