Nonlinear Waves, Optics, and Coupled Oscillators

该项目支持NSF工业博士后研究员Deborah Alterman在普林斯顿大学和新泽西州默里山朗讯-贝尔实验室的应用和计算数学项目中联合开展研究。教师导师将菲利普·霍姆斯，力学和应用数学教授，和朗讯科技的主管将博士迈克尔·温斯坦谁最近加入了数学科学研究中心（基础数学系）。我们设想两个主要的研究领域：（1）应用于光通信问题的非线性波动理论，和（2）作为神经网络模型的耦合振荡器。要使用和开发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，非线性波和稳定性理论。 一个普遍的主题是减少动力系统与无限或高维相空间，描述分布的现象，更简单的“振幅”偏微分或“模态”常微分方程。这种简化的模型可以提高对基础物理的理解，并为预测和设计提供捷径。 该GOALI项目由MPS多学科活动办公室（OMA）和数学科学部（DMS）联合支持。

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).