| 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
|
