MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Diffractive geometric optics for Bloch wave packets

Gregoire Allaire, Mariapia Palombaro and Jeffrey Rauch


We study, for times of order $1/\e$, solutions of wave equations which are $\Op(\e^2)$ modulations of an $\e$ periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order $\e$. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on times scales $t=\Op(1)$. A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields $H^1$ asymptotics when the envelopes are only $H^1$.

MSC Codes:
35B40, 35B27, 35L30, 35B34, 35J10
geometric optics, diffractive geometric optics, Bloch waves, diffraction, homogenization