Higher-order integrable models for oceanic internal wave–current interactions

In this paper, we derive a higher-order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat-surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long-wave, and small-amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.