MiS Preprint Repository

We study, for times of order $1/\text{e}$, solutions of wave equations which are $\text{Op}({\text{e}}^2)$ modulations of an $\text{e}$ periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order $\text{e}$. 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 $t=\text{Op}(1)$. 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 $H^1$.