Quasilinear evolution and periodic orbits in Hamiltonian systems

本研究将增进对有限维和无限维哈密顿系统的理解。本文将研究哈密顿型非线性色散波动方程中有点反直觉的拟线性演化，并给出拟线性行为的一般判据。研究将从具体的方程开始，如费米-帕斯塔-乌拉姆系统、Korteweg-ed Vries方程、Majda-McLaughlin-Tabak模型等。对于高频初始数据，由于色散的微妙平均效应，非线性动力学变为线性。除了纯粹的非线性动力学兴趣之外，理解拟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中周期轨道集的结构。该方法是基于几何控制理论和外部微分系统的方法。期望能证明大（双参数）族周期轨道的可解析性。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的价值。特别是，周期轨道的零概率存在意味着Dirichlet问题中特征值的Weyl渐近精度很高。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。光通信系统提供了最有效的长距离数据传输方式。信息通常通过短时间的光脉冲传输。在理想的情况下，传入的数据流在传输线的另一端不会失真，因此信息不会损坏。实际上，有两种主要的影响会使光脉冲失真：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，非线性迫使振幅较大的波与振幅较小的波的移动方式不同。非线性引起的失真尤其不可取。最近，光学系统工程师发现了一个显著的事实，即通过将脉冲封装得越来越紧密，非线性效应会变得越来越小。在早期的研究中？拟线性？对一个简单的模型问题，通过证明它在某些精确表述的条件下发生来解释现象。拟议研究的一部分旨在制定其他重要系统的准线性行为的一般标准，例如浅水动力学，固体物理和水波。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，其研究对基本系统，如经典的台球问题具有重要意义。在这里，我们应该考虑一个椭圆形的空腔，其中光线（或质点）沿直线传播，在边界处受到反射。我们相信，给定的射线产生闭合（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，我们提出要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.