Nonlinear Waves, Optics, and Coupled Oscillators

该项目支持NSF工业博士后Deborah Alterman在普林斯顿大学和新泽西州默里希尔的朗讯-贝尔实验室开展应用和计算数学项目的联合研究。学院导师将是力学和应用数学教授菲利普·霍姆斯，朗讯技术公司的主管将是最近加入数学科学研究中心(基础数学系)的迈克尔·韦恩斯坦博士。我们设想了两个主要的研究领域：(1)将非线性波动理论应用于光通信问题，(2)将耦合振子作为神经网络的模型。将使用和发展的数学工具包括动力系统和分叉理论、常微分方程和偏微分方程、渐近性以及非线性波动和稳定性理论。一个普遍的主题是将描述分布现象的具有无限或高维相空间的动力系统简化为更简单的“振幅”偏微分方程组或“模态型”常微分方程式。这种简化的模型有助于更好地理解潜在的物理原理，并为预测和设计提供了捷径。该GOALI项目由MPS多学科活动办公室(OMA)和数学科学处(DMS)共同支持。

This project supports an NSF-Industrial Postdoctoral Fellow, Deborah Alterman, to carry out research jointly in the Program in Applied and Computational Mathematics at Princeton University and Lucent-Bell Laboratories, Murray Hill, NJ. The Faculty Mentor will be Philip Holmes, Professor of Mechanics and Applied Mathematics, and the Supervisor at Lucent Technologies will be Dr Michael Weinstein who has recently joined the Mathematical Sciences Research Center (Fundamental Mathematics Department). We envisage two principal areas of research: (1) Nonlinear wave theory applied to problems of optical communications, and (2) Coupled oscillators as models of neural networks. The mathematical tools to be used and developed include dynamical systems and bifurcation theory, ordinary and partial differential equations, asymptotics, and nonlinear wave and stability theory. A pervasive theme is the reduction of dynamical systems with infinite or high dimensional phase spaces, describing distributed phenomena, to simpler ``amplitude'' partial differential or ``modal'' ordinary differential equations. Such reduced models lead to improved understanding of the underlying physics, and offer shortcuts in predictions and design. This GOALI project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).