October  2012, 5(5): 879-901. doi: 10.3934/dcdss.2012.5.879

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, United States

2. 

Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany, Germany, Germany

Received  January 2011 Revised  June 2011 Published  January 2012

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.
Citation: Martina Chirilus-Bruckner, Christopher Chong, Oskar Prill, Guido Schneider. Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 879-901. doi: 10.3934/dcdss.2012.5.879
References:
[1]

J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373.  doi: 10.1007/s00205-005-0378-1.  Google Scholar

[2]

K. Busch, G. Schneider, L. Tkeshelashvili and H. Uecker, Justification of the nonlinear Schrödinger equation in spatially periodic media,, Z. Angew. Math. Phys., 57 (2006), 905.  doi: 10.1007/s00033-006-0057-6.  Google Scholar

[3]

F. Chazel, Influence of bottom topography on long water waves,, M2AN Math. Model. Numer. Anal., 41 (2007), 771.  doi: 10.1051/m2an:2007041.  Google Scholar

[4]

M. Chirilus-Bruckner, "Nonlinear Interaction of Pulses,", Ph.D. Thesis, (2009).   Google Scholar

[5]

C. Chong and G. Schneider, The validity of the KdV-approximation in case of resonances arising from periodic media,, J. Math. Anal. Appl., 383 (2011), 330.  doi: 10.1016/j.jmaa.2011.05.028.  Google Scholar

[6]

W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits,, Comm. Partial Differential Equations, 10 (1985), 787.   Google Scholar

[7]

W.-P. Düll and G. Schneider, Justification of the Nonlinear Schrödinger equation for a resonant Boussinesq model,, Indiana Univ. Math. J., 55 (2006), 1813.  doi: 10.1512/iumj.2006.55.2824.  Google Scholar

[8]

J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities,, Nonlinearity, 17 (2004), 551.  doi: 10.1088/0951-7715/17/2/011.  Google Scholar

[9]

J. Giannoulis and A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials,, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 493.  doi: 10.3934/dcdsb.2006.6.493.  Google Scholar

[10]

E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems,, Technical Report I., (1955).   Google Scholar

[11]

G. H. Golub and C. F. Van Loan, "Matrix Computations," Third edition,, Johns Hopkins Studies in the Mathematical Sciences, (1996).   Google Scholar

[12]

T. Iguchi, A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves,, Kyushu J. Math., 60 (2006), 267.  doi: 10.2206/kyushujm.60.267.  Google Scholar

[13]

A. Iserles, "A First Course in the Numerical Analysis of Differential Equations," Second edition,, Cambridge Texts in Applied Mathematics, (2009).   Google Scholar

[14]

G. James and P. Noble, Breathers on diatomic Fermi-Pasta-Ulam lattices,, Physica D, 196 (2004), 124.  doi: 10.1016/j.physd.2004.05.005.  Google Scholar

[15]

L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium,, Math. USSR Sbornik Surveys, 60 (1988), 457.  doi: 10.1070/SM1988v060n02ABEH003181.  Google Scholar

[16]

P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities,, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85.  doi: 10.1017/S0308210500020989.  Google Scholar

[17]

G. Schneider, Validity and limitation of the Newell-Whitehead equation,, Math. Nachr., 176 (1995), 249.  doi: 10.1002/mana.19951760118.  Google Scholar

[18]

G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms,, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69.  doi: 10.1007/s000300050034.  Google Scholar

[19]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model,, in, (2000), 390.   Google Scholar

[20]

Guido Schneider and C. Eugene Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension,, Comm. Pure Appl. Math., 53 (2000), 1475.  doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.  Google Scholar

[21]

Guido Schneider and C. Eugene Wayne, The rigorous approximation of long-wavelength capillary-gravity waves,, Arch. Ration. Mech. Anal., 162 (2002), 247.  doi: 10.1007/s002050200190.  Google Scholar

[22]

G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances,, Journal of Differential Equations, 216 (2005), 354.  doi: 10.1016/j.jde.2005.04.018.  Google Scholar

[23]

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

[24]

N. J. Zabusky and M. D. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states,, Phys. Rev. Lett., 15 (1965), 240.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

show all references

References:
[1]

J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373.  doi: 10.1007/s00205-005-0378-1.  Google Scholar

[2]

K. Busch, G. Schneider, L. Tkeshelashvili and H. Uecker, Justification of the nonlinear Schrödinger equation in spatially periodic media,, Z. Angew. Math. Phys., 57 (2006), 905.  doi: 10.1007/s00033-006-0057-6.  Google Scholar

[3]

F. Chazel, Influence of bottom topography on long water waves,, M2AN Math. Model. Numer. Anal., 41 (2007), 771.  doi: 10.1051/m2an:2007041.  Google Scholar

[4]

M. Chirilus-Bruckner, "Nonlinear Interaction of Pulses,", Ph.D. Thesis, (2009).   Google Scholar

[5]

C. Chong and G. Schneider, The validity of the KdV-approximation in case of resonances arising from periodic media,, J. Math. Anal. Appl., 383 (2011), 330.  doi: 10.1016/j.jmaa.2011.05.028.  Google Scholar

[6]

W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits,, Comm. Partial Differential Equations, 10 (1985), 787.   Google Scholar

[7]

W.-P. Düll and G. Schneider, Justification of the Nonlinear Schrödinger equation for a resonant Boussinesq model,, Indiana Univ. Math. J., 55 (2006), 1813.  doi: 10.1512/iumj.2006.55.2824.  Google Scholar

[8]

J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities,, Nonlinearity, 17 (2004), 551.  doi: 10.1088/0951-7715/17/2/011.  Google Scholar

[9]

J. Giannoulis and A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials,, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 493.  doi: 10.3934/dcdsb.2006.6.493.  Google Scholar

[10]

E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems,, Technical Report I., (1955).   Google Scholar

[11]

G. H. Golub and C. F. Van Loan, "Matrix Computations," Third edition,, Johns Hopkins Studies in the Mathematical Sciences, (1996).   Google Scholar

[12]

T. Iguchi, A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves,, Kyushu J. Math., 60 (2006), 267.  doi: 10.2206/kyushujm.60.267.  Google Scholar

[13]

A. Iserles, "A First Course in the Numerical Analysis of Differential Equations," Second edition,, Cambridge Texts in Applied Mathematics, (2009).   Google Scholar

[14]

G. James and P. Noble, Breathers on diatomic Fermi-Pasta-Ulam lattices,, Physica D, 196 (2004), 124.  doi: 10.1016/j.physd.2004.05.005.  Google Scholar

[15]

L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium,, Math. USSR Sbornik Surveys, 60 (1988), 457.  doi: 10.1070/SM1988v060n02ABEH003181.  Google Scholar

[16]

P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities,, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85.  doi: 10.1017/S0308210500020989.  Google Scholar

[17]

G. Schneider, Validity and limitation of the Newell-Whitehead equation,, Math. Nachr., 176 (1995), 249.  doi: 10.1002/mana.19951760118.  Google Scholar

[18]

G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms,, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69.  doi: 10.1007/s000300050034.  Google Scholar

[19]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model,, in, (2000), 390.   Google Scholar

[20]

Guido Schneider and C. Eugene Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension,, Comm. Pure Appl. Math., 53 (2000), 1475.  doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.  Google Scholar

[21]

Guido Schneider and C. Eugene Wayne, The rigorous approximation of long-wavelength capillary-gravity waves,, Arch. Ration. Mech. Anal., 162 (2002), 247.  doi: 10.1007/s002050200190.  Google Scholar

[22]

G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances,, Journal of Differential Equations, 216 (2005), 354.  doi: 10.1016/j.jde.2005.04.018.  Google Scholar

[23]

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

[24]

N. J. Zabusky and M. D. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states,, Phys. Rev. Lett., 15 (1965), 240.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

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