No abstract

A (2+1)-dimensional rotating shallow water system with an underlying circular paraboloidal bottom topography is shown to admit a multiparameter integrable nonlinear subsystem of Ermakov-Ray-Reid type. The latter system, which describes the time evolution of the semi-axes of the elliptical moving shoreline on the paraboidal basin, is also Hamiltonian. The complete solution of the generic eight-dimensional dynamical system governing the reduction is obtained in terms of an elliptic integral representation.

A hydrodynamics-type system incorporating a Madelung-Bohm-type quantum potential, as derived by Wagner et al via Maxwell's equations and the paraxial approximation in nonlinear optics, is reduced to a nonlinear Schrödinger canonical form. A two-parameter nonlinear Ermakov-Ray-Reid system that arises from this model, and which governs the evolution of beam radii in an elliptically polarised medium is shown to be reducible to a classical Pöschl-Teller equation. A class of exact solutions to the Ermakov-type system is constructed in terms of elliptic dn functions. It is established that integrable twocomponent Ermakov-Ray-Reid subsystems likewise arise in a coupled (2+1)dimensional nonlinear optics model descriptive of the two-pulse interaction in a Kerr medium. The Hamiltonian structure of these subsystems allows their complete integration.