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Mathematics Colloquium


Date:  Tuesday, January 13, 2009

Title:  Linear and Nonlinear Resonance

Abstract:  Two problems which arise in energy conserving/Hamiltonian and spatially extended (non- compact) systems governed by PDEs are: 1) The dynamics of a coherent structure (e.g. an optical or matter wave "soliton" moving in a nonlinear medium, a gas bubble deforming in a fluid) interacting with other coherent structures or with an non-homogeneous environment; and 2) The long-time confinement of energy, e.g. for optical storage in a region of space and in a pre- ferred mode. Both problems can be understood in terms of resonant energy transfer among subsystems: one with discrete degrees of freedom ("oscillators") and one with a continuum of degrees of freedom ("fields"). In linear problems, resonances are characterized via a time-independent non-self ad- joint spectral problem, or as poles of an analytically continued resolvent operator. Howev- er, nonlinear resonance phenomena must be understood via time-dependent nonlinear scattering and dynamical systems methods. We give an introduction to analytical work and applications.

Speaker:  Michael Weinstein
Institution:  Columbia University


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