Infinite Dimensional Dynamical Systems and Partial Differential Equations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。韦恩教授将研究无限维动力系统的行为，如费米-帕斯塔-乌拉姆模型、纳维-斯托克斯方程和欧拉方程。他将使用动力系统理论的方法对这些系统的解进行定性和定量预测，并将集中在四个主要领域：(i)无限维色散哈密顿系统中孤波解的稳定性和相互作用，以及这些系统相空间的几何形状；（ii）流体表面上波包演化的近似方程的推导和证明，以及这些结果与线性部分具有连续谱的哈密顿系统的正规形式定理的关系；（iii）近无粘Navier-Stokes方程和其他弱耗散系统的亚稳行为；（iv）用不变流形定理分析奇摄动偏微分方程。不变流形等物体的几何性质在阐明有限维动力系统的行为方面有很大的帮助，本项目将发展类似的方法和对无限维系统行为的见解，特别是那些定义在无界空间区域的线性问题具有连续谱的系统。韦恩教授将要研究的微分方程出现在各种不同的物理环境中，其特点是，尽管这些方程本身众所周知，但它们太复杂了，除非在特殊和/或不现实的情况下才能求解。尽管如此，应用程序至少需要对其解的行为进行定性理解，本项目将对上述列举的方程进行这样的理解。作为与上一段第(i)点相关的一个例子，描述海洋波浪的方程对于理解气候和天气以及预测海啸等事件都很重要，它有一组解被称为“孤立波”，代表穿越海洋的单个波。然而，在实践中，许多波不可避免地存在，因此有必要了解这些波如何相互作用。这个项目将研究在这样的系统中可能发生的相互作用的类型及其后果。该研究项目的其余三个部分将以类似的方式，将对具有物理重要性的方程的特殊或极限情况的理解扩展到更现实的条件。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Professor Wayne will study the behavior of infinite dimensional dynamical systems such as the Fermi-Pasta-Ulam model, the Navier-Stokes equations and the Euler equations. He will use methods from dynamical systems theory to make qualitative and quantitative predictions about the solutions of these systems and will focus on four main areas: (i) The stability and interactions of solitary wave solutions in infinite dimensional dispersive Hamiltonian systems, and the geometry of the phase space of such systems; (ii) The derivation and justification of approximate equations for the evolution of wave packets on a fluid surface and the relation of such results to normal form theorems for Hamiltonian systems in which the linear part has continuous spectrum; (iii) Metastable behavior in the nearly inviscid Navier-Stokes equations and other weakly dissipative systems; and (iv) The use of invariant manifold theorems to analyze singularly perturbed partial differential equations. The geometrical properties of objects like invariant manifolds have been a great aid in illuminating the behavior of finite dimensional dynamical systems and this project will develop similar methods and insights into the behavior of infinite dimensional systems, particularly those defined on unbounded spatial regions where the linear problem has continuous spectrum.The differential equations that Professor Wayne will study arise in a variety of different physical circumstances and are characterized by the fact that while the equations themselves are well known they are too complicated to solve except in special and/or unrealistic cases. Nonetheless, applications require at least a qualitative understanding of the behavior of their solutions and this project will develop such an understanding for the equations enumerated above. As an example related to point (i) in the preceding paragraph, the equations that describe waves on the ocean, which are of importance both for understanding climate and weather and for predicting events such as tsunamis, have a family of solutions known as ``solitary waves'' which represent a single wave traveling across theocean. In practice, however, many waves are inevitably present and it becomes necessary to understand how these waves interact with each other. This project will study the types of interactions that can occur in such systems and their consequences. The remaining three sections of the research project will aim, in a similar fashion, to extend the understanding of special or limiting cases of equations of physical importance to more realistic conditions.