Diffractive Geometric Optics for Bloch Wave Packets
Abstract
We study, for times of order 1/ε, solutions of wave equations which are O(ε 2 ) modulations of an ε periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order ε. 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 = O(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 .
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BibTeX
@article{allaire2026diffractive,
title = {{Diffractive Geometric Optics for Bloch Wave Packets}},
author = {Grégoire Allaire and Mariapia Palombaro and Jeffrey Rauch},
journal = {Archive for Rational Mechanics and Analysis},
year = {2026},
month = sep,
doi = {10.1007/s00205-011-0452-9},
url = {https://hal.science/hal-05767699v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Diffractive Geometric Optics for Bloch Wave Packets
AU - Grégoire Allaire
AU - Mariapia Palombaro
AU - Jeffrey Rauch
PY - 2026
DA - 2026/09/28
JO - Archive for Rational Mechanics and Analysis
DO - 10.1007/s00205-011-0452-9
UR - https://hal.science/hal-05767699v1
AB - We study, for times of order 1/ε, solutions of wave equations which are O(ε 2 ) modulations of an ε periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order ε. 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 = O(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 .
SN - 0003-9527
SN - 1432-0673
ER -