Dynamics of Nonlinear Wave Equations

世界由波动方程支配：电路板上的电流，光纤电缆中的光，甚至银河系中心的黑洞都是通过波动动力学传播的。虽然波动方程无处不在，但人们对它的理解还远远不够。这个项目的目标是了解面对波自身或其环境的干扰时，波的动力学如何变化。这项研究试图了解一些海浪何时以及为什么消散，而另一些海浪持续存在，还有一些海浪坍塌。了解波在关键条件下的行为推动了技术进步--更小的微芯片，更快的数据传输，以及对宇宙形成的更深层次的洞察。这个项目将在几何环境中研究非线性波动和色散方程的解的动力学性质，在几何环境中，非线性结构与诸如曲率等经典几何概念密切相关。这样的方程出现在无数的物理模型中，从爱因斯坦的广义相对论方程到核物理中的粒子相互作用。具体的例子包括波图系统，杨-米尔斯系统，半线性版本的Skyrme方程，以及功率类型的非线性波和薛定谔方程。在许多这样的问题中，孤子(相干孤立波)被认为是全球时间动力学的基本构件：随着解的演变，它分解成有限数量的弱相互作用的孤子加上表现出线性动力学的余项。这是孤子分辨猜想的松散表述，而孤子分辨猜想在该领域中具有重要意义。在证明孤子分辨率方面的进展是本项目要解决的几个具体问题的根本动机。孤子的动力学也是奇点形成的基础，通常起着通过集中质量或能量形成奇点的普遍轮廓的作用。该项目将研究奇点形成的精细机制，并将试图从从集中点辐射出来的能量分布中表征溶液可能的动力学性质。

The world is ruled by wave equations: the electricity on a circuit board, the light in fiber-optic cables, and even the black hole in the center of the galaxy propagate by wave dynamics. Though ubiquitous, wave equations are far from understood. The goal of this project is to understand how the dynamics of waves change in the face of interference with themselves or with their environment. The research seeks to learn when and why some waves disperse, while other waves persist, and still others collapse. Knowing how waves behave in critical conditions drives technological progress -- smaller microchips, faster data transmission, and deeper insights into the formation of the universe. This project will study dynamical properties of solutions to nonlinear wave and dispersive equations in geometric settings where the nonlinear structure is intimately tied to classical geometric notions such as curvature. Such equations arise in a myriad of physical models, ranging from Einstein's equations of general relativity, to the interactions of particles in nuclear physics. Particular examples of interest include the wave maps systems, the Yang-Mills system, semilinear versions of Skyrme's equation, and power-type nonlinear wave and Schrodinger equations. In many of these problems, solitons (coherent solitary waves) are thought to be the basic building blocks of global-in-time dynamics: as the solution evolves, it decomposes into a finite number of weakly interacting solitons plus a remainder term exhibiting linear dynamics. This is a loose formulation of the soliton resolution conjecture, which is of central importance in the field. Progress towards the proof of soliton resolution is the underlying motivation of several of the specific problems to be addressed in this project. The dynamics of solitons are also fundamental to singularity formation, often playing the role of the universal profiles for singularities that form via a concentration of mass or energy. The project will study the fine mechanics of singularity formation, and will seek to characterize the possible dynamical properties of the solution from the profile of the energy that has radiated away from the point of concentration.