Homogenization theory predicts that waves in periodic heterogeneous media can be described, in the limit of small periodicity, by an effective wave equation. This is true as long as finite observation times are considered, but it is no longer true for large time intervals, which are relevant in many applications. On large time intervals, one observes dispersion, which means that waves of different wave-length travel with different speed. In particular, a wave pulse will, in general, change its form in the course of time. A linear wave equation with constant coefficients does not show dispersion and cannot explain the observed effect. We must instead find a dispersive model. We showed that, effectively, a linear wave equation with periodic coefficients and with a small periodicity can be replaced, in a new homogenization limit, by a linear wave equation of fourth order with constant coefficients. The predictions of this weakly dispersive model agrees perfectly with numerical results. We furthermore investigate the wave equation in a discrete spring-mass model. The discrete character of the model introduces small-scale oscillations, which result again in a dispersive long time behavior. We derive the dispersive effective wave equations also for the discrete model. Moreover, for ring-like solution fronts that occur for localized initial data after long time, we derive the equations that dictate the evolution of the front: Our derivation provides a linearized KdV equation and an explicit representation of the corresponding initial data in Fourier space. We present work that was obtained in collaborations with A. Lamacz, T. Dohnal, and F. Theil.

Einladung von Prof. Jens Rademacher