Simple two-layer dispersive models in the Hamiltonian reduction formalism

A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of Benjamin (1986 J. Fluid Mech. 165 445-74) for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac’s theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids.