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

Martina Chirilus-Bruckner; Christopher Chong; Oskar Prill; Guido Schneider

doi:10.3934/dcdss.2012.5.879

Article Contents

2012, Volume 5, Issue 5: 879-901. Doi: 10.3934/dcdss.2012.5.879

This issue Previous Article Preface Next Article Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation

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

1.
Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215
2.
Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart

Received: December 31, 2010

Revised: May 31, 2011

Published: September 2012

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Abstract

It is the purpose of this paper to prove error estimates for the approximate description of macroscopic wave packets in infinite 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 discrete Bloch wave transform of the underlying infinite-dimensional system of coupled ODEs. After this transform the existing proof for the associated approximation theorem for the NLS approximation used for the approximate description of oscillating wave packets in dispersive PDE systems transfers almost line for line. In contrast, the proof of the approximation theorem for the KdV approximation of long waves is less obvious. In a special situation we prove a first approximation result.

Keywords:

KdV,
NLS,
polyatomic FPU,
approximation.

Mathematics Subject Classification: Primary: 34E20; Secondary: 35Q53, 35Q55.

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