Nonlinear Waves and Stability in Partial Differential Equations

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系统中创建波动现象。提出的研究重点是发展和改进分析几种物理系统中波的稳定性的方法。这些包括1)非线性色散介质中的孤波，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部非径向解；3)液汽临界点附近流体的内波。正在开发的方法涉及改进使用：a)埃文斯函数来分析具有对称性的二维特征值问题；B)解算算子的奇异摄动理论，用于研究晶格动力学和水波下可积系统的稳定性结果如何在不可积系统中持续存在；C)不适定系统中的无限维中心流形理论，研究三维中行水波的存在性；d)低速流动的零马赫数渐近分析，以研究临界点附近的水动力现象，特别是：在强分层平衡附近的内波阻尼率，以及近临界流体单相流动中可能的毛细效应。本研究第一部分的总体目标是了解非线性波动现象的“鲁棒性”。在罕见的，所谓的“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，所以我们需要知道哪些现象依赖于可积性，哪些不依赖于可积性。一项重要的基础设施技术，其中非线性波是重要的，并没有完全了解是长距离通信通过光纤。第二部分工作的动机是在航天飞机上进行的物理实验；此外，超临界流体在材料加工中的应用也越来越广泛和发展。这种流体在临界点附近的流动行为是不寻常的，人们对其知之甚少，这在过去的实验设计中导致了代价高昂的失败。需要进行基础调查，以建立关于这些流的知识库，这些流可以作为应用程序开发的基础。

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.