Nonlinear Waves and Stability in Partial Differential Equations

非线性波和偏微分方程的稳定性

• 批准号: 9704924
• 负责人: Robert Pego
• 金额: $ 12.01万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704924&HistoricalAwards=false
• 关键词: Nonlinear; Waves; Stability; Partial; Differential

9704924 Pego Nonlinearity helps to create wave phenomena in a variety of important systems of partial differential equations that arise in science and engineering. The focus of the proposed research is on developing and improving methods for analyzing the stability of waves in several systems of physical interest. These include 1) Solitary waves in nonlinear dispersive media, including lattice dynamics and water waves; 2) A recently discovered class of localized nonradial solutions of nonlinear Schrodinger equations in 2+1 dimensions; and 3) Internal waves in fluids near the liquid-vapor critical point. The methods under development involve improving the use of: a) Evans functions to analyze eigenvalue problems in two dimensions with symmetries; b) singular perturbation theory for resolvent operators, to study how stability results for integrable systems persist in nonintegrable systems for lattice dynamics and water waves; c) infinite-dimensional center manifold theory in ill-posed systems, to study the existence of traveling water waves in three dimensions; d) zero-Mach-number asymptotic analysis of low-velocity flows, to study hydrodynamic phenomena near the critical point, specifically: damping rates of internal waves about a strongly stratified equilibrium, and possible capillary effects in one-phase flows of near-critical fluids. The general goal of the first part of this research is to understand how "robust" are nonlinear wave phenomena. Nonlinear waves in rare, so-called "integrable" systems can be very well understood due to what seems miraculous -- they can be solved in closed form. But most realistic systems are not integrable, so one needs to know what phenomena depend on integrability and what do not. An important infrastructural technology where nonlinear waves are important and are not completely understood is long-distance communication via optical fiber. The second part of the work was motivated by physics experiments carried out on the spa ce shuttle; also, the use of supercritical fluids in materials processing is extensive and growing. The behavior of flows of such fluids near the critical point is unusual and little understood, and this has led to costly failures in experimental design in the past. Fundamental investigations are needed to build the knowledge base about such flows that can serve as the foundation for the development of applications.

9704924 PEGO非线性有助于在科学和工程中出现的各种重要的部分微分方程系统中创建波浪现象。拟议的研究的重点是开发和改进方法，用于分析几种物理感兴趣系统中波的稳定性。其中包括1）非线性分散介质中的孤立波，包括晶格动力学和水波； 2）在2+1维中，最近发现的非线性Schrodinger方程的局部非放射溶液； 3）液体震荡临界点附近的流体中的内波。 所开发的方法涉及改进以下方法：a）Evans函数在与对称性的两个维度中分析特征值问题； b）分析运算符的奇异扰动理论，以研究集成系统的稳定性如何持续存在于晶格动力学和水波的不可整合系统中； c）在不足的系统中，无限维中心歧管理论，以研究三个维度的行进水波的存在； d）对低速流量的零射线渐近分析，以研究临界点附近的流体动力现象，特别是：在强烈分层平衡的内部波的阻尼速率，以及在近临界流体的单相流中可能的毛细管作用。 这项研究的第一部分的一般目标是了解非线性波现象的“健壮”是如何的。由于看起来奇迹般的奇迹，可以很好地理解稀有，所谓的“可集成”系统中的非线性波 - 可以以封闭形式解决。但是，大多数现实的系统都不可集成，因此需要知道哪些现象取决于整合性以及什么不取决于。非线性波很重要并且尚不完全了解的重要基础设施技术是通过光纤进行长途通信。工作的第二部分是由在Spa Ce Shuttle上进行的物理实验激励的。同样，在材料加工中使用超临界流体是广泛且增长的。临界点附近的这种流体流动的行为是不寻常的，几乎没有理解，这导致了过去实验设计的昂贵故障。 需要进行基本的调查来建立有关可以作为应用程序开发的基础的知识基础。