<b>MCTP - Math Colloquium</b><br><i>Linear and Nonlinear Resonance</i>

Speaker: Michael I. Weinstein (Columbia University)

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 preferred 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 adjoint spectral problem, or as poles of an analytically continued resolvent operator. However, nonlinear resonance phenomena must be understood via time-dependent nonlinear scattering and dynamical systems methods. We give an introduction to analytical work and applications.