We derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free interface coupled with a free surface, this latter situation occurring more commonly in experiment and in nature. Based on the linearized equations, we highlight the discrepancies between the cases of rigid lid and free surface upper boundary conditions, which in some circumstances can be significant. We also derive systems of nonlinear dispersive long wave equations in the large amplitude regime, and compute solitary wave solutions of these equations.

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