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：电路板上的电力，光纤电缆中的光，甚至是银河系中心的黑洞传播了波动力学。尽管无处不在，但波动方程远非理解。该项目的目的是了解在干扰自己或环境中，海浪动态如何变化。该研究试图学习何时以及为什么有些海浪散布，而其他海浪持续存在，而另一些波则崩溃了。知道波浪在临界条件下的行为如何推动技术进步 - 较小的微芯片，更快的数据传输以及对宇宙形成的更深入的见解。该项目将研究在几何环境中非线性波和分散方程的解决方案的动态特性，在几何环境中，非线性结构与经典的几何概念（例如曲率）密切相关。这种方程式出现在无数的物理模型中，范围从爱因斯坦的一般相对论到核物理学中粒子的相互作用。感兴趣的特定示例包括波浪映射系统，杨利尔斯系统，Skyrme方程的半线性版本以及功率类型的非线性波和Schrodinger方程。在许多问题中，孤子（连贯的孤立波）被认为是全球动力学的基本构件：随着解决方案的发展，它将其分解为有限数量的弱相互作用的孤子，其余的术语表现出线性动力学。这是孤子分辨率猜想的松散公式，这在田间至关重要。朝索证明解决方案的进展是该项目要解决的几个特定问题的基本动机。孤子的动力学也是奇异性形成的基础，通常会扮演通过质量或能量浓度形成的奇异性的通用曲线的作用。该项目将研究奇异性形成的精细力学，并将试图从从浓度点辐射的能量的曲线来表征溶液的可能动力学特性。