Dynamics of Nonlinear Wave Equations

世界是由波动方程所支配的：电路板上的电流、光纤电缆中的光，甚至是银河系中心的黑洞都是通过波动力学传播的。虽然波动方程无处不在，但人们对它的理解还很遥远。该项目的目标是了解波浪的动力学如何在面对自身或环境干扰时发生变化。这项研究试图了解一些波何时以及为什么会分散，而另一些波则会持续存在，还有一些波会崩溃。了解波在临界条件下的行为将推动技术进步-更小的微芯片，更快的数据传输以及对宇宙形成的更深入了解。这个项目将研究非线性波和色散方程在几何设置中的解的动力学性质，其中非线性结构与经典几何概念（如曲率）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。感兴趣的具体例子包括波映射系统、杨-米尔斯系统、Skyrme方程的半线性版本以及幂型非线性波和薛定谔方程。在许多这些问题中，孤子（相干孤立波）被认为是全球时间动力学的基本组成部分：随着解决方案的发展，它分解为有限数量的弱相互作用孤子加上表现出线性动力学的余项。这是一个松散的制定孤子决议猜想，这是在该领域的核心重要性。孤子分辨率的证明进展是本项目中要解决的几个具体问题的潜在动机。孤子的动力学也是奇点形成的基础，通常扮演着通过质量或能量集中形成的奇点的普适轮廓的角色。该项目将研究奇点形成的精细机制，并将设法从从集中点向外辐射的能量分布图中确定解决方案可能的动力学性质。

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.