Quasilinear evolution and periodic orbits in Hamiltonian systems

这一研究将增进对有限和无限维哈密顿系统的理解。我们将研究哈密顿型非线性色散波动方程的一种有违直觉的拟线性演化，并发展出拟线性行为的一般判据。研究将从具体的方程开始，如Fermi-Pasta-Ulam系统，Korteweg-ed Vries方程，Majda-McLaughlin-Tabak模型等。由于色散的微妙平均效应，对于高频初值，非线性动力学变为线性。除了纯粹的非线性动力学兴趣，理解准线性现象对于工程系统设计也很重要，因为设计线性系统更容易，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。该方法基于几何控制理论和外微分系统的方法。期望证明大的(两参数)周期轨道族的解析性。虽然周期轨道集的结构是理论动力学中的一个重要问题，但对其他科学领域也具有重要的价值。特别地，周期轨道的零概率出现意味着Dirichlet问题本征值的Weyl渐近性具有很高的精度。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作用。光通信系统提供了最有效的远距离数据传输方式。信息通常通过短持续时间的光脉冲来传输。在理想情况下，传入的数据流在传输线的另一端看起来不会失真，因此信息不会被破坏。在现实中，有两个主要效应会使光脉冲失真：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，而非线性迫使幅度较大的波与幅度较小的波移动方式不同。由非线性引起的失真尤其令人不快。最近，光学系统工程师发现了一个显著的事实，即通过越来越紧密地包装脉冲，非线性效应变得更小。在早期的工作中，这是什么？准线性？对于一个简单的模型问题，通过证明它在某些精确的公式条件下发生，解释了该现象。拟议研究的一部分旨在制定其他重要系统的拟线性行为的一般标准，如浅水动力学、固态物理和水波。该项目的第二部分涉及力学中周期轨道的研究。周期轨道产生循环行为，其研究对于经典台球问题等基本系统具有重要意义。在这里，人们应该想到一个椭圆形腔，光线(或点质量)沿直线传播，在边界处遭受反射。人们认为，给定射线将产生闭合(即，周期性)轨迹的概率为零。这个问题与其他领域有着深刻的联系，建议理解这个概率何时为零，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.