Mathematical Sciences: Fundamental Aspects of Nonlinear Dynamics

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This project will cover three basic interrelated areas of applied mathematics. The first is concerned with large amplitude motions both quasi periodic and almost periodic in dynamical systems of many degrees of freedom. The principal investigator will develop new methods of study, going beyond the KAM theory for small perturbations, that utilize a novel nonlinear p.d.e. approach. The second related topic involves vortex dynamics of an ideal inviscid fluid. Here the principal investigator will research the outstanding problems of leapfrogging of vortex rings, classes of vortex cores, and vortex breakdown of bubble and spiral type. Finally, the principal investigator will study of "flux phenomenon" in the new theories of high Tc superconductivity, an area of prime importance in high technology. All these areas require deep understanding of nonlinear science and its associated mathematical structures. This has been the central focus of the investigator's research for over two decades.