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Dynamics of Nonlinear Wave Equations

非线性波动方程的动力学

基本信息

批准号：
1700127
负责人：
Andrew Lawrie
金额：
$ 15万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700127&HistoricalAwards=false
关键词：
Dynamics Nonlinear Wave Equations

项目摘要

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.

世界是由波动方程统治的：电路板上的电、光纤电缆中的光，甚至星系中心的黑洞都是通过波动动力学传播的。尽管波动方程无处不在，但人们还远未理解它。该项目的目标是了解波浪的动力学在面对自身或环境的干扰时如何变化。该研究旨在了解一些波何时以及为何消散，而另一些波持续存在，还有一些波崩溃。了解波在关键条件下的行为可以推动技术进步——更小的微芯片、更快的数据传输以及对宇宙形成的更深入的了解。该项目将研究几何设置中非线性波和色散方程解的动力学特性，其中非线性结构与曲率等经典几何概念密切相关。这些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粒子的相互作用。感兴趣的具体示例包括波图系统、Yang-Mills 系统、Skyrme 方程的半线性版本以及功率型非线性波和薛定谔方程。在许多此类问题中，孤子（相干孤波）被认为是全局时间动力学的基本构建块：随着解的发展，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。这是孤子分辨率猜想的松散表述，在该领域具有核心重要性。孤子解析证明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，通常起着通过质量或能量集中形成的奇点的通用轮廓的作用。该项目将研究奇点形成的精细力学，并将寻求从从集中点辐射出的能量分布来表征解决方案可能的动力学特性。