Nonlinear Waves, Optics, and Coupled Oscillators

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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).