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.

DMS-1311553的摘要Wayne将研究诸如不变的歧管理论和Kolmogorov-Arnold-Moser（KAM）理论之类的方法，这些方法最初是为了理解有限的尺寸动力学系统而开发的，这些方法可以适应有限的尺寸动力学系统，以洞悉部分差分公式的定性和定量行为。他将主要集中于在物理应用中产生的方程式，例如基本流体方程，例如Navier-Stokes方程，涡流纸的方程以及来自非线性光学的方程。 他的研究将集中在四个主要问题上：（a）二维流体中的亚稳定行为； （b）涡旋方程的定期解决方案； （c）定期媒体中的呼吸； （d）分散性哈密顿系统中的正常形式和不变流形。 动力学系统方法对没有封闭形式解决方案的有限维系统的定性行为产生了许多见解。 现在已经建立了许多要研究的无限尺寸系统的存在理论，这项研究将尝试得出有关解决方案在各种物理相关的时间尺度上的行为的更详细的信息，并照亮这些时间量表的起源。明确除非非常特殊或身体不切实际。 然而，应用至少需要对其解决方案行为的定性理解，该研究项目将旨在对上述系统发展这样的理解。 作为与上一段​​中点（C）相关的示例，请考虑当前用于电信的光纤电缆中使用的类型的光脉冲。 在诸如玻璃之类的均匀介质中，脉冲迅速散布或分散。 但是，在定期培养基中，像某些晶体一样，这种脉冲可能会被困。 由于希望它们可以作为纯粹的光学计算系统的基础，因此被困的脉冲具有极大的当前兴趣。韦恩教授的研究将研究媒介必须满足的条件，以支持这些``被困''脉冲，并研究这种媒体的普遍性。 其他三个项目还将旨在发展对这些重要物理系统的新基本见解。