Diffractive Geometric Optics for Bloch Wave Packets

We study, for times of order 1/ε, solutions of wave equations which are 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 the 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 the small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales. A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H¹ asymptotics when the envelopes are only H¹.