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.
Nonlinear dispersive equations model a variety of physical processes. A few particular examples include light propagation in nonlinear media, waves in plasma, water waves on the surface of a lake, etc. Dispersion generally broadens wave packets (or wave trains) and nonlinearity may broaden waves in spectral domain. In some applications, the presence of nonlinearity may be desirable, as e.g. in optical communication systems, dispersion and nonlinearity balance each other to produce a stable solitary wave that could be used as information carrier. In other applications, nonlinearity might be detrimental, and it is desirable to suppress it. The major goal of this project was to understand the mechanism which could reduce nonlinear effects in nonlinear dispersive equations. One way to do it is to evolve small initial data, so nonlinearity is very weak, while the dispersive effects are the same. However, in many applications the signal must be sufficiently large to keep the signal-to-noise ratio bounded. Then, a more subtle mechanism of nonlinearity suppression is needed. Such mechanism was established in this project: restrict the initial data to the high frequency part of the spectrum. It turns out that nonlinearity gets averaged out by the high frequency components and effectively linear evolution takes place. The main outcome of this project was to develop a general methodology to study such nearly linear dynamics and give quantitative estimates when and in what sense such evolution takes place.
