Dynamical Systems Methods for Partial Differential Equations

偏微分方程的动力系统方法

• 批准号: 1311553
• 负责人: Clarence Wayne
• 金额: $ 59.95万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-10-01 至 2019-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311553&HistoricalAwards=false
• 关键词: Dynamical; Systems; Methods; Partial; Differential

Abstract for DMS-1311553Professor Wayne will study how methods such as invariant manifold theory and the Kolmogorov-Arnold-Moser (KAM) theory, which were originally developed to understand finite dimensional dynamical systems, can be adapted to yield insight into the qualitative and quantitative behavior of solutions of partial differential equations. He will concentrate primarily on equations arising in physical applications such as fundamental fluid equations, e.g the Navier-Stokes equation, the equations for vortex sheets, and equations from nonlinear optics. His research will focus on four main problems: (A) Metastable behavior in two-dimensional fluids; (B) Periodic solutions of the vortex sheet equations; (C) Breathers in periodic media; and (D) Normal forms and invariant manifolds in dispersive Hamiltonian systems. Dynamical systems methods have yielded many insights into the qualitative behavior of finite dimensional systems for which no closed form solutions exist. The existence theory of many of the infinite dimensional systems to be studied is now well established and this research will attempt to derive more detailed information about the behavior of the solutions on various physically relevant time scales, as well as illuminating the origin of these time scales.The differential equations that Professor Wayne will study arise in a variety of different physical contexts and are characterized by the fact that while the equations themselves are well known, they are too complicated to solve explicitly except in very special or physically unrealistic cases. Nevertheless, applications require at least a qualitative understanding of the behavior of their solutions and this research project will aim to develop such an understanding for the systems described above. As an example related to point (C) in the preceding paragraph, consider light pulses of the types that are used in fiber optic cables currently used for telecommunications. In a homogeneous medium, such as glass, such pulses rapidly spread out or disperse. However, in periodic media, like certain crystals, such pulses may become trapped. Trapped pulses are of great current interest because of the hope that they might serve as the basis for a purely optical computational system. Professor Wayne's research will examine the conditions that the medium must satisfy in order to support these ``trapped'' pulses as well as investigating how common such media are likely to be. The other three projects will also aim to develop new fundamental insights into these important physical systems.

摘要DMS-1311553韦恩教授将研究如何方法，如不变流形理论和Kolmogorov-Arnold-Moser（KAM）理论，这最初是为了理解有限维动力系统，可以适应产生洞察力的定性和定量行为的解决方案的偏微分方程。他将主要集中在物理应用中产生的方程，如基本流体方程，例如Navier-Stokes方程，涡面方程和非线性光学方程。 他的研究将集中在四个主要问题：（A）亚稳态行为在二维流体;（B）涡面方程的周期解;（C）呼吸周期性介质;和（D）正常形式和不变流形在色散哈密顿系统。 动力系统方法已经产生了许多见解的定性行为的有限维系统，不存在封闭形式的解决方案。 许多待研究的无穷维系统的存在性理论现在已经很好地建立起来，本研究将试图获得关于各种物理相关时间尺度上的解的行为的更详细的信息，以及阐明这些时间尺度的起源。韦恩教授将研究的微分方程出现在各种不同的物理背景下，其特点是，这些方程本身是众所周知的，它们太复杂了，除非在非常特殊或物理上不现实的情况下，否则无法显式求解。 然而，应用程序需要至少有一个定性的了解他们的解决方案的行为，这个研究项目的目的是发展这样的理解上述系统。 作为与前段中的点（C）相关的示例，考虑在当前用于电信的光纤电缆中使用的类型的光脉冲。 在均匀介质中，如玻璃，这种脉冲迅速扩散或分散。 然而，在周期性介质中，如某些晶体，这种脉冲可能会被捕获。 捕获的脉冲是目前的极大兴趣，因为希望他们可能作为一个纯粹的光学计算系统的基础。韦恩教授的研究将检查介质必须满足的条件，以支持这些“被捕获”的脉冲，以及调查如何共同这种媒体可能是。 其他三个项目也将致力于为这些重要的物理系统开发新的基本见解。