Asymptotic Methods for Singularity Perturbed Nonlinear Systems

奇异摄动非线性系统的渐近方法

• 批准号: 0207201
• 负责人: Alexander Tovbis
• 金额: $ 8.7万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0207201&HistoricalAwards=false
• 关键词: Asymptotic; Methods; Singularity; Perturbed; Nonlinear

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.

该项目的目的是推进分析工具的开发，以研究各种渐近和奇异摄动问题，包括聚焦非线性薛定谔方程（NLS）的半经典（或零色散）极限以及奇异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且无序振荡区域的形成。这与散焦 NLS 的情况形成鲜明对比，目前人们已经很好地理解了散焦 NLS 的情况。 黎曼-希尔伯特问题的方法将用于描述聚焦 NLS 的复杂行为。该项目的第二部分与奇扰动和离散系统中同宿和异宿连接的破坏现象有关。非线性光学的许多方面，例如在非线性介质中传播的光束的稳态轮廓以及在非线性介质中传播的某些脉冲光纤，可以通过 NLS 方程及其各种渐近状态来描述。 我们将集中讨论该方程最困难的渐近问题之一。该项目的第二部分研究动力系统中可积性和混沌之间的过渡现象。