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 \varepsi > 0, we derive the continuum limit equation for time scales of order \varepsi-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.

Representation of solutions to wave equations with profile functions

“…Our analysis can be understood as a continuation and improvement of [9], where the authors studied the long time behavior for a lattice wave equation. They derived, on the one hand, that a weakly dispersive wave equation in a homogeneous medium is a valid replacement for the lattice wave equation.…”

“…They derived, on the one hand, that a weakly dispersive wave equation in a homogeneous medium is a valid replacement for the lattice wave equation. On the other hand, [9] introduced the shell reconstruction operator; one result regards the approximate reconstruction of the solution from profiles that are obtained as solutions of a linearized KdV equation.…”

“…We finally introduce the shell reconstruction operator. The reconstruction was also used in [9]; it maps a family of profiles to a ring-like (d = 2) or shell like (d = 3) function on R d . An important ingredient is the rescaling factor (ct) −(d−1)/2 , which has the effect that L 2 -norms of reconstructed functions are bounded.…”

“…This is not a restriction. General initial data can be treated by an appropriate decomposition, see [9] for details. A profile z → V (z; q) with a direction parameter q ∈ S d−1 is used along the ray x = zq; the profile is centered in order to have the main pulse near {x ∈ R 2 : |x| = ct}.…”

Justification of the KP-II approximation in dynamics of two-dimensional FPU systems

“…Another work can be found in [32], where propagation of rings in two-dimensional lattices was analyzed in the linear approximation and compared rigorously with the approximation of the linearized KdV equation (rather than with the linearized KP-II equation).…”