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 nonhomogeneous environment; and
2) The longtime 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 timeindependent nonself ad
joint spectral problem, or as poles of an analytically continued resolvent operator. Howev
er, nonlinear resonance phenomena must be understood via timedependent nonlinear scattering
and dynamical systems methods. We give an introduction to analytical work and applications.
Speaker: Michael Weinstein
Institution: Columbia University
