On a Whitham-Broer-Kaup-like system arising in the oceanic shallow water

Abstract

Presently, taking care of the oceanic shallow water, people have investigated some Kaup-type systems. For example, a Whitham-Broer-Kaup-like system, which we aim to study hereby, models the dispersive long waves in the oceanic shallow water. As for the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, using symbolic computation, we build up (1) a scaling transformation, (2) two hetero-Bäcklund transformations, each of which is from that system to a known linear partial differential equation, and (3) a set of the similarity reductions, from that system to a known ordinary differential equation with some sample solutions also presented. Our results rely on different dispersion/diffusion powers for that system, with respect to the oceanic shallow water.