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偏微分方程组中创建波动现象。拟议研究的重点是开发和改进分析几个物理感兴趣的系统中波浪稳定性的方法。其中包括 1) 非线性色散介质中的孤立波，包括晶格动力学和水波； 2）最近发现的一类2+1维非线性薛定谔方程的局域非径向解； 3) 液体-蒸汽临界点附近流体中的内波。 正在开发的方法包括改进以下方法的使用： a) 埃文斯函数，用于分析具有对称性的二维特征值问题； b) 求解算子的奇异摄动理论，研究可积系统的稳定性结果如何在晶格动力学和水波的不可积系统中持续存在； c) 病态系统中的无限维中心流形理论，研究三维行进水波的存在性； d) 低速流的零马赫数渐近分析，研究临界点附近的流体动力学现象，具体来说：强分层平衡附近的内波阻尼率，以及近临界单相流中可能的毛细管效应液体。 本研究第一部分的总体目标是了解非线性波现象的“鲁棒性”。罕见的所谓“可积”系统中的非线性波可以很好地理解，因为看起来很神奇——它们可以以封闭形式求解。但大多数现实系统都是不可积的，因此我们需要知道哪些现象取决于可积性，哪些现象不依赖于可积性。非线性波很重要但尚未被完全理解的一项重要基础设施技术是通过光纤进行的长距离通信。这项工作的第二部分是受到航天飞机上进行的物理实验的启发。此外，超临界流体在材料加工中的应用正在广泛且不断增长。这种流体在临界点附近的流动行为是不寻常的，人们对此知之甚少，这在过去导致了实验设计中代价高昂的失败。 需要进行基础研究来建立有关此类流程的知识库，以作为应用程序开发的基础。