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.\ Kre{\u\i}n and G. Ja. Ljubarski{\u\i} [\textit{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 function 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, \textit{Slow Light in Photonic Crystals}, Waves Random Complex Media, 16 (2006), pp.\ 293--382].
