Integrable Ermakov structure in nonlinear continuum mechanics and optics

Abstract

Ermakov-Ray-Reid systems have recently attracted much attention due to their novel invariant of motion, nonlinear superposition principles and extensive physical applications. In this thesis, our main concern is with integrable structure underlying certain models in nonlinear continuum mechanics and optics via reduction to such Ermakov-type systems. The main contributions of this thesis are as follows : In hydrodynamics, a shallow water system with a circular paraboloidal bottom topography is investigated via the elliptic vortex procedure. Key theorems analogous to those of Ball and Cushman-Roisin et al are generalised and used to construct the analytical vortex solutions in terms of an elliptic integral function. In particular, a class of typical pulsrodon solutions with a breather-type free boundary oscillation is isolated and its behaviour is simulated. In nonlinear optics, a coupled 2+1-dimensional optics model is studied via a variational approach. Three distinct reductions to integrable Ermakov systems are set down. The underlying Hamiltonian structures render their complete integration. It is shown that integrable Hamiltonian Ermakov systems likewise arise in a 3+1-dimensional optics model. In particular, an Ovisannikov-Dyson type reduction is obtained wherein the eigenmode of the solution explains a remarkable flip-over effect observed experimentally. Integrable Ermakov-Ray-Reid structure is shown to arise out of a 2+1 dimensional modulated Madelung system with logarithmic and Bohm quantum potentials via an exponential-type elliptic vortex ansatz. In addition, exact analytical solutions of the original system are obtained in terms of an elliptic integral representation. In magnetogasdynamics, a power-type elliptic vortex ansatz and two-parameter pressure-density relation are introduced into a 2+1-dimensioanl magnetogasdynamic system and a finite dimensional nonlinear dynamical system is thereby obtained. The latter admits integrable Hamiltonian Ermakov structure and a Lax pair formulation when the adiabatic index γ = 2. Exact solutions of the magnetogasdynamic systems are constructed which describe a rotating elliptic plasma cylinder bounded by a vacuum state.