Colloquium: Tuesday, December 1, 2pm. Speaker: Jeremy Schi ff (Bar-Ilan). Title: Multiscale analysis of breathing-beating transitions.

We study certain 2 degree-of-freedom Hamiltonian systems arising from an
approximation scheme for solutions of the 2d nonlinear Schrodinger equation with cubic-quintic or
saturated nonlinearities (possibly in a grade-indexed medium). The solutions of these systems
can be of various different types, depending on the values of the parameters and the initial
conditions, with transitions – which we call \breathing-beating” transitions – associated with
solutions displaying extremely long time periodic behavior. We show that this behavior is typically
associated with a 1-1 Hamiltonian resonance, and use multiscale analysis to successfully
predict some, but certainly not all, of the transitions.
For those who maybe did not quite manage to understand all of the above: It is remarkable
fact that seemingly simple looking differential equations can have solutions that behave in
qualitatively different ways over substantially different scales. Multiscale analysis is a set of
tools for explaining such phenomena. I will try to explain one of the key tricks of multiscale
analysis (not assuming any previous familiarity) and its application to a problem of interest in
the field of nonlinear optics.
Joint work with David Ianetz.