Lattice dynamics on large time scales and dispersive effective equations

Abstract

We investigate the long time behavior of waves in crystals. Starting from a linear wave equation on a discrete lattice with periodicity ε > 0, we derive the continuum limit equation for time scales of order ε^(-2). The effective equation is a weakly dispersive wave equation of fourth order. Initial values with bounded support result in ring-like solutions and we characterize the dispersive long-time behavior of the radial profiles with a linearized KdV equation of third order.