Mathematical methods for nonlinear wave interactions

非线性波相互作用的数学方法

• 批准号: 0506495
• 负责人: Roy Goodman
• 金额: --
• 依托单位: New Jersey Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0506495&HistoricalAwards=false
• 关键词: Mathematical; methods; nonlinear; wave; interactions

Abstract, DMS 0506495, R Goodman, New Jersey Inst of technologyTitle: Mathematical methods for nonlinear wave interactions This project studies interactions between waves in models from optics and mathematical physics, and the relative stability of waves trapped by localized potentials. Wave interactions in many applications have been known for many years to display resonance structures in which wave capture and reflection alternate in a surprising fractal-like manner. Methods from the theory of dynamical systems, including Melnikov methods, matched asymptotics, and transport theory of iterated maps, are to be applied to ordinary differential equations models of such wave interactions, extending recently published work of the PI. In a second thread of research, we look at the nonlinear interaction between trapped modes at potentials engineered into optical fibers, modeled by the nonlinear Schrodinger equation (NLS), or the nonlinear coupled mode equations (NLCME) in the case of fiber Bragg gratings. The NLCME system lacks an energy-minimization principle (all bound states are saddle points of the energy), yet somehow a ground state is chosen. We study this using numerical analysis, derivation of simplified ordinary differential equation models, and ideas from the spectral theory of Hamiltonian systems.Nonlinear wave phenomena are ubiquitous in physics. An important engineering example is in optical communications, where information is sent as pulses of light through glass fibers. It is important to understand how these waves interact with each other, and with local structures in the medium through which they travel, in order to better design devices that exploit their novel properties. The aim of this research is to extend mathematical theory from dynamical systems to explain phenomena seen in wave interaction, such as wave-trapping. It will use sophisticated computational and analytical methods, some developed in the study of Bose-Einstein condensates, to study the behavior of light in novel physical configurations.

摘要，DMS 0506495，R Goodman，新泽西理工学院标题：非线性波相互作用的数学方法该项目研究光学和数学物理模型中波之间的相互作用，以及局域势捕获的波的相对稳定性。 多年来，许多应用中的波相互作用已知会显示共振结构，其中波捕获和反射以令人惊讶的类似分形的方式交替。动力系统理论的方法，包括 Melnikov 方法、匹配渐近法和迭代映射的传输理论，将应用于此类波相互作用的常微分方程模型，扩展了 PI 最近发表的工作。 在第二个研究方向中，我们研究了光纤中设计的电位处的俘获模式之间的非线性相互作用，通过非线性薛定谔方程（NLS）或光纤布拉格光栅情况下的非线性耦合模式方程（NLCME）进行建模。 NLCME 系统缺乏能量最小化原理（所有束缚态都是能量的鞍点），但以某种方式选择了基态。 我们使用数值分析、简化常微分方程模型的推导以及哈密顿系统谱理论的思想来研究这一问题。非线性波现象在物理学中无处不在。 一个重要的工程示例是光通信，其中信息通过玻璃纤维作为光脉冲发送。 重要的是要了解这些波如何相互作用，以及如何与它们传播的介质中的局部结构相互作用，以便更好地设计利用其新颖特性的设备。 这项研究的目的是将数学理论从动力系统扩展到解释波相互作用中看到的现象，例如波捕获。 它将使用复杂的计算和分析方法（其中一些方法是在玻色-爱因斯坦凝聚态研究中开发的）来研究光在新型物理配置中的行为。