Mathematics Graduate Student Colloquium

MGSC Website: http://math.uci.edu/~mgsc/

Perturbation Analysis of Slow Waves for Periodic Differential-Algebraic Equations of Definite Type

Aaron Welters
Wednesday, November 10, 2010
4:00 pm - 4:50 pm
RH 114

Talk Abstract:

In this talk we consider linear periodic differential-algebraic equations (DAEs) that depend analytically on a spectral parameter. In particular, we extend the results of M. G. Krein and G. Ja. Ljubarskii [Amer. Math. Soc. Transl. (2) Vol. 89 (1970), pp. 1--28] to linear periodic DAEs of definite type and study the analytic properties of Bloch waves and their Floquet multipliers as functions of the spectral parameter. Our main result is the connection between a non-diagonalizable Jordan normal form of the monodromy matrix for the reduced differential system associated with the DAEs and the occurrence of slow Bloch waves for the periodic DAEs, i.e., Bloch solutions of the periodic DAEs which propagate with near zero group velocity. We show that our results can be applied to the study of slow light in photonic crystals [A. Figotin and I. Vitebskiy, Slow Light in Photonic Crystals, Waves Random Complex Media, 16 (2006), pp. 293--382].

About the Speaker:

Aaron Welters is currently pursuing his Ph.D. in Mathematics from the University of California at Irvine (UCI). Under his advisor, Professor Alexander Figotin, he has been studying mathematical problems related to Spectral and Scattering Theory, Slow Light, and Photonic Crystals. He expects to defend his thesis and receive his Ph.D. in June 2011.

Refreshments:

Pizza will be served after the talk.