From: BrendaSueCook (brenda@math.missouri.edu)
Date: Fri Feb 23 2007 - 15:25:06 CST
COLLOQUIUM
University of Missouri-Columbia
Department of Mathematics
Gerald Teschl,
University of Vienna, Austria
Stability of the Periodic Toda Lattice under Short Range Perturbations
Abstract: Solitons are well-known to be the stable part of short-range
perturbations of completely integrable wave equations. So far, it is
generally believed that a perturbed periodic integrable system splits
asymptotically into a number of solitons plus a decaying radiation part,
a situation similar to that observed for perturbations of the constant
solution. In this lecture I want to demonstrate via numerical
experiments that this is not true; instead, the radiation part does not
decay, but manifests itself asymptotically as a modulation of the
periodic solution which undergoes a continuous phase transition in the
isospectral class of the periodic background solution.
To understand this phenomenon analytically, we will start with a
Riemann-Hilbert problem on a Riemann surface, integrate abelian
differentials, and fix some ambiguities in the Cauchy kernel using
Riemann theta functions. This allows us to deform the jump along the
spectrum into the complex plane until almost nothing is left. Finally,
we follow the steepest descent and perturb a singular integral equation.
This is based on joint work with Spyros Kamvissis.
3:30 p.m.
Thursday, March 1
114 General Classroom Building
Refreshments will be served at 3:00 p.m. in Room 326 Mathematical
Sciences (Math Lounge).
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