Asymptotic Methods for Singularity Perturbed Nonlinear Systems

该项目的目的是推进分析工具的发展，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）极限和奇摄动动力系统中的指数小效应。 聚焦NLS的数值实验揭示了剧烈和无序振荡区域的形成。这与散焦NLS的情况形成鲜明对比，散焦NLS现在已经被合理地理解了。 采用Riemann-Hilbert问题的方法来描述聚焦非线性光学系统的复杂行为。本项目的第二部分与奇摄动和离散化系统中同宿和异宿联系的破缺现象有关。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方程及其各种渐近状态来描述。 我们将集中讨论这个方程最困难的渐近问题之一。第二部分研究动力系统中可积性和混沌之间的过渡现象。

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.