Mathematical methods for nonlinear wave interactions

摘要，DMS 0506495，R古德曼，新泽西技术研究所标题：非线性波相互作用的数学方法本项目研究从光学和数学物理模型中的波之间的相互作用，和相对稳定性的波被困局部势。 多年来，人们已经知道在许多应用中，波的相互作用显示出共振结构，其中波的捕获和反射以令人惊讶的分形方式交替。从动力系统理论的方法，包括Melnikov方法，匹配的渐近性，和运输理论的迭代映射，被施加到常微分方程模型的这种波的相互作用，扩展最近发表的工作的PI。 在第二个线程的研究中，我们看看陷波模式之间的非线性相互作用的潜力工程光纤，建模的非线性薛定谔方程（NLS），或非线性耦合模方程（NLCME）的情况下，光纤布拉格光栅。 NLCME系统缺乏能量最小化原则（所有束缚态都是能量的鞍点），但不知何故选择了基态。 我们使用数值分析、简化的常微分方程模型推导以及哈密顿系统谱理论的思想来研究这一点。非线性波动现象在物理学中无处不在。 一个重要的工程例子是光通信，其中信息以光脉冲的形式通过玻璃纤维发送。 重要的是要了解这些波是如何相互作用的，以及如何与它们所穿过的介质中的局部结构相互作用，以便更好地设计利用其新特性的设备。 这项研究的目的是从动力系统中扩展数学理论，以解释波相互作用中出现的现象，如陷波。 它将使用复杂的计算和分析方法，其中一些是在玻色-爱因斯坦凝聚研究中开发的，以研究光在新物理配置中的行为。

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.