Mathematical Sciences: Coherent and Chaotic Phenomena in PDE's

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This project concerns equations whose evolution may be modelled on the interaction of coherent analytic structures which range from vortices to multi-phase or multi-similarity wave packets to cavitating and collapsing filaments to monopoles. The detailed structure of these objects, which can be determined from asymptotics or geometry or by using the analytical properties of integrable systems, provides a precise description of the mechanisms which give rise to complicated and often chaotic dynamics. In current and future work, as more modern geometric and analytical techniques are integrated into the program the studies will be extended to systems modelled on non-classical geometric structures, statistical ensembles of classical structures, spatially extended defect patterns, and topologically complicated states such as knotted filaments in fluid dynamics or monopoles in non-abelian gauge theories. The mathematics involved in these studies covers a broad spectrum, from the theory of integrable and almost integrable systems and rigorous analysis in pde's to the more intuitive ideas best understood in a geometrical setting. In many cases, specifically in the investigations in convective patterns and optics, real experiments are used as a guide to develop and check the validity of models. In all cases, each theoretical investigation is accompanied by extensive numerical experiments.