Nonlinear Dispersive Wave Dynamics

提出的工作的目的是考虑在非线性色散方程领域的一系列具有挑战性的问题。总的来说，目标是提高我们对类波现象背景下非线性相互作用的理解。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，麦克斯韦系统和陈-西蒙斯-薛定谔模型。虽然大多数工作将涉及非线性问题，但如果没有对潜在的线性色散动力学有很好的理解，就无法获得良好的非线性结果。这里的两个主要主题是(i)边值问题和（ii）弯曲背景上的全局时间色散。后一个主题主要面向系统，例如麦克斯韦系统和自旋二波方程；这与在广义相对论中的爱因斯坦方程背景下对史瓦西黑洞和克尔黑洞的推测稳定性的研究有关。在非线性方面，要探讨的一个关键问题是能量临界半线性色散方程的大数据解。第一阶段包括理解和分类全局散射解的可能障碍（如孤子、稳态、自相似解）。在没有这些障碍的情况下，目标是获得全局适定性和散射。另一个目标是很好地理解这些障碍物附近的动力学。拟线性色散方程的短期和长期动力学研究是本提案的另一个主题。有几种令人感兴趣的模型，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模型。虽然提出的工作主要是理论性的，但许多问题都源于物理理论，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提供一些启示。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，该项目涉及大量研究生（目前有6名）和博士后（项目第一年有5名，其中包括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、网络工具、会议报告和暑期学校广泛传播。

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.