Asymptotic Methods for Singularity Perturbed Nonlinear Systems

该项目的目的是促进分析工具的发展，以研究各种渐近和奇摄动问题，包括聚焦非线性薛定谔方程（NLS）的半经典（或零色散）极限和奇摄动系统中的指数小效应。聚焦NLS的数值实验揭示了剧烈振荡和无组织振荡区域的形成。这与散焦NLS的情况形成了鲜明的对比，后者现在已经相当好地理解了。本文将采用黎曼-希尔伯特问题的方法来描述聚焦NLS的复杂行为。第二部分研究了奇摄动和离散系统中同斜和异斜连接的断裂现象。非线性光学的许多方面，如光束在非线性介质中传播的平稳分布和某些脉冲在光纤中的传播，都可以用NLS方程及其各种渐近域来描述。我们将集中讨论这个方程最困难的渐近问题之一。项目的第二部分研究了动力系统中可积性与混沌之间的转换现象。

The aim of the project is to advance the development of analytic tools to study various asymptotic and singularly perturbed problems, including the semiclassical (or zero dispersion) limit of the focusing nonlinear Schroedinger equation (NLS) and exponentially small effects in singularly perturbed dynamical systems. Numerical experiments for the focusing NLS reveal the formation of regions of violent and disorganized oscillations. This is in drastic contrast to the case of the defocusing NLS, which is reasonably well understood by now. The method of Riemann-Hilbert problems will be used to describe the complicated behavior of the focusing NLS. The second part of the project is connected with the phenomenon of breaking of homoclinic and heteroclinic connections in singularly perturbed and discretized systems.A number of aspects of nonlinear optics, such as the stationary profiles of beams propagating in nonlinear media and propagation of certain pulses in optical fibers, can be described by the NLS equation and its various asymptotic regimes. We will concentrate on one of the most difficult asymptotic problems for this equation. The second part of the project investigates the phenomenon of transition between integrability and chaos in dynamical systems.