Mathematical Sciences: Coherent and Chaotic Phenomena in PDE's

这个项目涉及的方程的演变可能是 以相干分析结构的相互作用为模型， 范围从漩涡到多相或多相似波 包空化和崩溃的丝单极。 的 这些物体的详细结构，可以从 渐近或几何或通过使用的分析性质 可积系统，提供了一个精确的描述 这些机制导致了复杂的、往往是混乱的 动力学 在当前和未来的工作中，随着更多的现代几何 和分析技术被整合到该计划中， 研究将扩展到非经典系统建模， 几何结构，经典的统计系综 结构，空间上扩展的缺陷图案，以及拓扑结构， 复杂的状态，如流体动力学中的打结细丝， 非阿贝尔规范理论中的单极子 数学 参与这些研究的人员涵盖了广泛的范围，从 可积和几乎可积系统的理论和严格 分析在偏微分方程的更直观的想法，最好的理解， 几何学的背景 在许多情况下，特别是在 对流模式和光学研究，真实的 实验被用来作为指导，以开发和检查的有效性 的模型。 在所有情况下，每个理论研究都是 伴随着大量的数值实验。

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.