Justification of leading order quasicontinuum approximations of strongly nonlinear lattices

Christopher Chong; P.G. Kevrekidis; Guido Schneider

doi:10.3934/dcds.2014.34.3403

Article Contents

2014, Volume 34, Issue 9: 3403-3418. Doi: 10.3934/dcds.2014.34.3403

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Justification of leading order quasicontinuum approximations of strongly nonlinear lattices

1.
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305
2.
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9315
3.
Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart

Received: February 28, 2013

Revised: October 31, 2013

Published: August 2014

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Abstract

We consider the leading order quasicontinuum limits of a one-dimensional granular medium governed by the Hertz contact law under precompression. The approximate model which is derived in this limit is justified by establishing asymptotic bounds for the error with the help of energy estimates. The continuum model predicts the development of shock waves, which are also studied in the full system with the aid of numerical simulations. We also show that existing results concerning the Nonlinear Schrödinger (NLS) and Korteweg de-Vries (KdV) approximation of FPU models apply directly to a precompressed granular medium in the weakly nonlinear regime.

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Mathematics Subject Classification: Primary: 35Q70, 34K07; Secondary: 35L67.

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