Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations

Abstract

It is the purpose of this paper to prove error estimatesfor the approximate description of macroscopic wave packets ininfinite periodic chains of coupled oscillators by modulation equations,like the Korteweg--de Vries (KdV) or the Nonlinear Schrödinger (NLS)equation. The proofs are based on a discreteBloch wave transform of the underlying infinite-dimensional system of coupled ODEs.After this transform the existing proof for the associatedapproximation theorem for the NLS approximation used forthe approximate description of oscillating wave packets in dispersive PDE systemstransfers almost line for line. In contrast, the proof of the approximation theoremfor the KdV approximation of long waves is less obvious. In a special situationwe prove a first approximation result.