Homogenization, and Bifurcation of Localized from Extended States

Thursday, May 22, 2014 - 10:20am - 11:00am
Keller 3-180
Michael Weinstein (Columbia University)
I will discuss recent results on the bifurcation of spatially localized states
from a continuum of extended states. This phenomenon plays an important role
in the mathematical study of wave propagation in ordered microstructures,
which are perturbed by spatially compact or non-compact defects.

Near the bifurcation point, there is strong spatial scale separation
and one expects the natural homogenized equation to govern.

Our first example is one in which this intuition does not apply, and an appropriate
effective equation must be derived. (Joint work with V. Duchene and I. Vukicevic)

Our second example concerns the bifurcation of
“topologically protected edge states” in a class of periodic structures,
perturbed by a (non-compact) domain wall. Such states play a
central role in many recently studied systems in condensed matter
physics and photonics. (Joint work with C.L. Fefferman and J.P. Lee-Thorp)
MSC Code: