Advanced Search
Article Contents
Article Contents

Justification of leading order quasicontinuum approximations of strongly nonlinear lattices

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35Q70, 34K07; Secondary: 35L67.


    \begin{equation} \\ \end{equation}
  • [1]

    K. Ahnert and A. Pikovsky, Compactons and chaos in strongly nonlinear lattices, Phys. Rev. E, 79 (2009), 026209.doi: 10.1103/PhysRevE.79.026209.


    M. Chirilus-Bruckner, C. Chong, O. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901.doi: 10.3934/dcdss.2012.5.879.


    J. M. English and R. L. Pego, On the solitary wave pulse in a chain of beads, Proceedings of the AMS, 133 (2005), 1763-1768 (electronic).doi: 10.1090/S0002-9939-05-07851-2.


    G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161 (1994), 391-418.doi: 10.1007/BF02099784.


    J. Goodman and P. Lax, On dispersive difference schemes, Comm. Pure. Appl. Math., 41 (1988), 591-613.doi: 10.1002/cpa.3160410506.


    E. B. Herbold and V. F. Nesterenko, Shock wave structure in a strongly nonlinear lattice with viscous dissipation, Phys. Rev. E, 75 (2007), 021304.doi: 10.1103/PhysRevE.75.021304.


    M. Herrmann, Oscillatory waves in discrete scalar conservation laws, Math. Models Methods Appl. Sci., 22 (2012), 1150002.doi: 10.1142/S021820251200585X.


    M. Herrmann and J. D. M Rademacher, Riemann solvers and undercompressive shocks of convex FPU chains, Nonlinearity, 23 (2010), 277-304.doi: 10.1088/0951-7715/23/2/004.


    H. Hertz, On the contact of elastic solids, J. Reine Angew. Math., 92 (1881), 156-171.


    T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.doi: 10.1007/BF00280740.


    P. G. Kevrekidis, Non-linear waves in lattices: Past, present, future, IMA Journal of Applied Mathematics, 76 (2011), 389-423.doi: 10.1093/imamat/hxr015.


    P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness, Phys. Rev. E, 65 (2002), 046613.doi: 10.1103/PhysRevE.65.046613.


    P. D. Lax, On dispersive difference schemes, Physica D, 18 (1986), 250-254.doi: 10.1016/0167-2789(86)90185-5.


    R. J. Leveque, Numerical Methods for Conservation Laws (Lectures in Mathematics), Birkhauser, 1992.doi: 10.1007/978-3-0348-8629-1.


    A. Molinari and C. Daraio, Stationary shocks in periodic highly nonlinear granular chains, Phys. Rev. E, 80 (2009), 056602.doi: 10.1103/PhysRevE.80.056602.


    V.F. Nesterenko, Dynamics of Heterogeneous Materials, Springer-Verlag, New York, 2001.doi: 10.1007/978-1-4757-3524-6.


    P. Rosenau, Dynamics of nonlinear mass-spring chains near the continuum limit, Physics Letters A, 118 (1986), 222-227.doi: 10.1016/0375-9601(86)90170-2.


    P. Rosenau, Dynamics of dense lattices, Phys. Rev. B, 36 (1987), 5868-5876.doi: 10.1103/PhysRevB.36.5868.


    G. Schneider, Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam system, Appl. Anal., 89 (2010), 1523-1539.doi: 10.1080/00036810903277150.


    G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, International Conference on Differential Equations. Proceedings of the Conference, Equadiff '99, Berlin, Germany, (ed. B. Fiedler et al.), (1999). 1.Singapore: World Scientific. 390-404, (2000).


    S. Sen, J. Hong, J. Bang, E. Avalos and R. Doney, Solitary waves in the granular chain, Physics Reports, 462 (2008), 21-66.doi: 10.1016/j.physrep.2007.10.007.


    J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, Heidelberg, Berlin, 1983.


    A. Stefanov and P. Kevrekidis, On the existence of solitary traveling waves for generalized hertzian chains, J. Nonlin. Sci., 22 (2012), 327-349.doi: 10.1007/s00332-011-9119-9.


    A. Stefanov and P. Kevrekidis, Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-564.doi: 10.1088/0951-7715/26/2/539.

  • 加载中

Article Metrics

HTML views() PDF downloads(80) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint