Quasilinear evolution and periodic orbits in Hamiltonian systems

这一研究将提高对有限维和无限维哈密顿系统的理解。本文研究了Hamilton型非线性色散波方程的一个有点违反直觉的拟线性演化，并给出了拟线性行为的一般判据。研究将开始与特定的方程，如费米-帕斯塔-乌拉姆系统，Korteweg-ed弗里斯方程，Majda-McLaughlin-Tabak模型等的非线性动力学成为线性的高频初始数据，因为一个微妙的平均效应，由于分散。 除了纯粹的非线性动力学的兴趣，了解准线性现象是重要的工程系统设计，因为设计线性系统更容易的原因有很多。 第二个研究领域是台球型哈密顿系统的周期轨道集的结构。该方法是基于几何控制理论和外微分系统的方法。 预计将证明周期轨道的大（两个参数）家庭的解析性。作为理论动力学中的一个重要问题，周期轨道集的结构对其他科学领域具有重要价值。特别是，零概率出现的周期轨道意味着高精度的外尔渐近的特征值的狄利克雷问题。 在物理学中，周期轨道集的结构在量子混沌和光学微腔的研究中起着重要的作用。光通信系统提供了长距离数据传输的最有效方式。信息通常是通过短持续时间的光脉冲传输的。 在理想情况下，输入的数据流在传输线的另一端不会失真，因此信息不会被破坏。实际上，两个主要的效应使光脉冲失真：色散和非线性。 色散表征了波的速度在多大程度上取决于频率，而非线性则迫使振幅较大的波与振幅较小的波以不同的方式运动。由于非线性引起的失真是特别不希望的。最近，光学系统工程师们发现了一个显著的事实，即通过越来越紧密地包装脉冲，非线性效应变得越来越小。 在早期的工作中，这？拟线性？现象解释了一个简单的模型问题，证明它发生在某些精确制定的条件下。拟议的研究的一部分，旨在开发在其他重要的系统，如浅水动力学，固态物理和水波准线性行为的一般标准。该项目的第二部分涉及力学中周期轨道的研究。 周期轨道产生常返性，对它的研究对于基本系统（如经典台球问题）具有重要意义。这里我们可以想象一个椭圆形的空腔，光线（或质点）在其中沿着直线传播，在边界处受到反射。 据信给定射线产生闭合（即，周期性的）轨迹。这个问题与其他领域有着深刻的联系，我们建议了解这种概率何时为零，何时为正。

This study will improve the understanding of finite and infinite dimensional Hamiltonian systems. A somewhat counter-intuitive quasilinear evolution in nonlinear dispersive wave equations of Hamiltonian type will be investigated and a general criterion of quasilinear behavior developed. The study will begin with specific equations such as the Fermi-Pasta-Ulam system, the Korteweg-ed Vries equation, the Majda-McLaughlin-Tabak model, etc. The nonlinear dynamics becomes linear for high frequency initial data because of a subtle averaging effect due to the dispersion. Besides purely nonlinear dynamics interest, understanding the quasilinear phenomenon is important for engineering systems design, since designing linear systems is easier for a number of reasons. A second area of study is the structure of the set of periodic orbits in Hamiltonian systems of billiard type. The approach is based on the methods of geometric control theory and exterior differential systems. It is expected that analyticity of the large (two parameter) families of periodic orbits will be proved. While being an important problem in theoretical dynamics, the structure of the set of periodic orbits is of great value to other areas of science. In particular, zero probability occurrence of periodic orbits implies high accuracy of Weyl's asymptotics for eigenvalues in the Dirichlet problem. In physics, the structure of the set of periodic orbits plays an important role in the study of quantum chaos and in optical microcavities. Optical communication systems provide the most effective way of data transmission over long distances. The information is usually transmitted by light pulses of short duration. In an ideal world, the incoming data stream would appear undistorted at the other end of the transmission line, so the information would not be corrupted. In reality, two major effects distort the light pulses: dispersion and nonlinearity. Dispersion characterizes how much the speed of the wave depends on the frequency, and nonlinearity forces larger amplitude waves to move differently from the smaller amplitude ones. The distortion due to nonlineariy is especially undesirable. Recently, optical system engineers have discovered the remarkable fact that by packing pulses more and more tightly, nonlinear effects become smaller. In earlier work this ?quasilinear? phenomenon was explained for a simple model problem by demonstrating that it occurs under certain precisely formulated conditions. One part of the proposed research aims to develop general criteria of quasilinear behavior in other important systems, such as shallow water dynamics, solid state physics, and water waves. The second part of the project deals with the study of periodic orbits in mechanics. Periodic orbits produce recurrent behavior and their study is of great importance for basic systems, such as the classical billiard problem. Here one should think of an oval cavity in which a light ray (or a point mass) travels along straight lines, suffering reflections at the boundary. It is believed that the probability is zero that a given ray will produce a closed (i.e., periodic) trajectory. This issue has deep connections to other fields, and it is proposed to understand when this probability is zero and when it is positive.