Mathematical Sciences: Coherent and Chaotic Phenomena in PDE's

该项目涉及的方程的演化可能以相干分析结构的相互作用为模型，这些结构的范围从涡旋到多相或多相似波包，到空化和坍缩细丝到单极子。这些物体的详细结构，可以从渐近性或几何或利用可积系统的解析性质来确定，提供了产生复杂且通常是混沌动力学的机制的精确描述。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，研究将扩展到以非经典几何结构为模型的系统，经典结构的统计集成，空间扩展缺陷模式，以及拓扑复杂的状态，如流体力学中的结丝或非阿贝尔规范理论中的单极子。这些研究涉及的数学涵盖了广泛的范围，从可积和几乎可积系统的理论和pde的严格分析到更直观的思想，最好在几何设置中理解。在许多情况下，特别是在对流模式和光学的研究中，实际实验被用作指导来开发和检查模型的有效性。在所有情况下，每一个理论研究都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.