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September  2014, 34(9): 3403-3418. doi: 10.3934/dcds.2014.34.3403

Justification of leading order quasicontinuum approximations of strongly nonlinear lattices

Christopher Chong 1, , P.G. Kevrekidis 2, and  Guido Schneider 3,

1. 

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305, United States
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, Germany

Received  March 2013 Revised  November 2013 Published  March 2014

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.
Keywords: error estimates, strongly nonlinear, granular crystals, Quasicontinuum approximation, shocks..
Mathematics Subject Classification: Primary: 35Q70, 34K07; Secondary: 35L6.
Citation: Christopher Chong, P.G. Kevrekidis, Guido Schneider. Justification of leading order quasicontinuum approximations of strongly nonlinear lattices. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3403-3418. doi: 10.3934/dcds.2014.34.3403
References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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

[15]

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.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

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

[20]

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).  Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

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.  Google Scholar

[24]

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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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

[15]

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.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

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

[20]

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).  Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

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.  Google Scholar

[24]

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.  Google Scholar

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