Nonlinear Dispersive Wave Dynamics

所提出的工作的目的是考虑广泛的非线性色散方程领域内的挑战性问题。 从广义上讲，我们的目标是提高我们的非线性相互作用的背景下，波的现象的理解。感兴趣的方程包括波动方程，薛定谔方程和KdV型方程，以及物理相关的耦合系统，如爱因斯坦方程，水波，麦克斯韦系统和Chern-Simons-Schroedinger模型。虽然大部分的工作将涉及非线性问题，一个人不能获得良好的非线性结果，没有一个很好的理解潜在的线性色散动力学。 这里的两个主要主题是（i）边值问题和（ii）弯曲背景上的全球时间色散。后一个主题主要面向系统，例如麦克斯韦系统和自旋两波方程;这与在广义相对论的爱因斯坦方程的背景下研究Schwarzchild和Kerr黑洞的约束稳定性有关。在非线性方面，要探索的一个关键主题是能量临界半线性色散方程的大数据解。第一阶段涉及理解和分类可能的障碍物（如孤子，稳态，自相似的解决方案），全球散射的解决方案。 在没有这种障碍的情况下，目标是获得全球适定性和散射。另一个目标是实现这些障碍物附近的动力学的一个很好的理解。短期和长期动力学的准线性色散方程的研究是本建议的另一个主题。 有几个模型的兴趣，即非线性波动方程（特别是爱因斯坦方程），非线性薛定谔方程以及一些水波models.Although拟议的工作主要是理论，许多问题有其起源的物理理论，如广义相对论，多体量子场论，表面波传播和等离子体物理。因此，希望研究结果可以对相应的物理现象有所启发。这方面的两个很好的例子是广义相对论中的黑洞稳定性问题，以及水波的长距离传播。在人力资源方面，该项目涉及大量的研究生（目前有6名），以及博士后（项目第一年有5名，包括3名NSF博士后）。它还涉及国内外各机构的许多合作者。这项研究的结果将通过出版物、基于网络的工具、会议和暑期学校的演讲广泛传播。

The aim of the proposed work is to consider a wide array of challenging problems within the field of nonlinear dispersive equations. Broadly speaking, the goal is to improve our understanding of nonlinear interactions in the context of wave-like phenomena. Equations of interest include wave equations, the Schroedinger equations and KdV type equations, as well as physically relevant coupled systems such as Einstein's equations, water waves, Maxwell systems and the Chern-Simons-Schroedinger model.While most of the work will be concerned with nonlinear problems, one cannot obtain good nonlinear results without having a good understanding of the underlying linear dispersive dynamics. The two main themes here are (i) boundary value problems and (ii) global in time dispersion on curved backgrounds. The latter topic is mainly oriented toward systems, e.g. the Maxwell system and the spin two wave equation; this is related to the study of the conjectured stability of the Schwarzchild and Kerr black holes in the context of the Einstein equations in general relativity.On the nonlinear side, a key topic to be explored is that of large data solutions in energy critical semilinear dispersive equations. The first stage involves understanding and classifying possible obstructions (e.g. solitons, steady states, self-similar solutions) to global scattering solutions. In the absence of such obstructions the goal is to obtain global well-posedness and scattering. Another goal is to achieve a good understanding of the dynamics near these obstructions.The study of short and long term dynamics in quasilinear dispersive equations is another main theme of this proposal. There are several models of interest, namely nonlinear wave equations (in particular Einstein's equations), nonlinear Schroedinger equations as well as some water wave models.Although the proposed work is primarily theoretical, many of the problems have their origin in physical theories such as general relativity, many body quantum field theory, surface wave propagation and plasma physics. As such, it is hoped that the results of the research may shed some light on the corresponding physical phenomena. Two good examples in this directions are the black hole stability question in general relativity, as well as the long range propagation of water waves.On the human resource side, the project involves a good number of graduate students (six at present), as well as postdocs (five for the first year of the project, including three NSF postdocs). It also involves many collaborators at various institutions, both domestic and abroad. The results of this research will be made widely available via publications, web based tools, presentations at conferences and summer schools.