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

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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

Martina Chirilus-Bruckner ^1, , Christopher Chong ^2, , Oskar Prill ^2, and Guido Schneider ^2,

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

Abstract
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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: approximation., NLS, KdV, polyatomic FPU.

Mathematics Subject Classification: Primary: 34E20; Secondary: 35Q53, 35Q5.

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 and 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), 373-410. doi: 10.1007/s00205-005-0378-1.
[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), 905-939. doi: 10.1007/s00033-006-0057-6.
[3]	F. Chazel, Influence of bottom topography on long water waves, M2AN Math. Model. Numer. Anal., 41 (2007), 771-799. doi: 10.1051/m2an:2007041.
[4]	M. Chirilus-Bruckner, "Nonlinear Interaction of Pulses," Ph.D. Thesis, Universität Karlsruhe, 2009. Available from: http://digbib.ubka.uni-karlsruhe.de/volltexte/1000012894.
[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), 330-336. doi: 10.1016/j.jmaa.2011.05.028.
[6]	W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003.
[7]	W.-P. Düll and G. Schneider, Justification of the Nonlinear Schrödinger equation for a resonant Boussinesq model, Indiana Univ. Math. J., 55 (2006), 1813-1834. doi: 10.1512/iumj.2006.55.2824.
[8]	J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities, Nonlinearity, 17 (2004), 551-565. doi: 10.1088/0951-7715/17/2/011.
[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), 493-523. doi: 10.3934/dcdsb.2006.6.493.
[10]	E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, Technical Report I., Los Alamos Rep, LA1940, Los Alamos, 1955; Reproduced in "Nonlinear Wave Motion" (ed. A. C. Newell), AMS, Providence, RI, 1974.
[11]	G. H. Golub and C. F. Van Loan, "Matrix Computations," Third edition, Johns Hopkins Studies in the Mathematical Sciences, John Hopkins University Press, Baltimore, MD, 1996.
[12]	T. Iguchi, A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves, Kyushu J. Math., 60 (2006), 267-303. doi: 10.2206/kyushujm.60.267.
[13]	A. Iserles, "A First Course in the Numerical Analysis of Differential Equations," Second edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.
[14]	G. James and P. Noble, Breathers on diatomic Fermi-Pasta-Ulam lattices, Physica D, 196 (2004), 124-171. doi: 10.1016/j.physd.2004.05.005.
[15]	L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium, Math. USSR Sbornik Surveys, 60 (1988), 457-483. doi: 10.1070/SM1988v060n02ABEH003181.
[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), 85-91. doi: 10.1017/S0308210500020989.
[17]	G. Schneider, Validity and limitation of the Newell-Whitehead equation, Math. Nachr., 176 (1995), 249-263. doi: 10.1002/mana.19951760118.
[18]	G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69-82. doi: 10.1007/s000300050034.
[19]	G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in "International Conference on Differential Equations," Vol. 1, 2 (Berlin, 1999), World Scientific Publ., River Edge, NJ, (2000), 390-404.
[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), 1475-1535. doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.
[21]	Guido Schneider and C. Eugene Wayne, The rigorous approximation of long-wavelength capillary-gravity waves, Arch. Ration. Mech. Anal., 162 (2002), 247-285. doi: 10.1007/s002050200190.
[22]	G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances, Journal of Differential Equations, 216 (2005), 354-386. doi: 10.1016/j.jde.2005.04.018.
[23]	G. Schneider, Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam-system, Applicable Analysis, 89 (2010), 1523-1539. doi: 10.1080/00036810903277150.
[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), 240-243. doi: 10.1103/PhysRevLett.15.240.

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), 373-410. doi: 10.1007/s00205-005-0378-1.
[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), 905-939. doi: 10.1007/s00033-006-0057-6.
[3]	F. Chazel, Influence of bottom topography on long water waves, M2AN Math. Model. Numer. Anal., 41 (2007), 771-799. doi: 10.1051/m2an:2007041.
[4]	M. Chirilus-Bruckner, "Nonlinear Interaction of Pulses," Ph.D. Thesis, Universität Karlsruhe, 2009. Available from: http://digbib.ubka.uni-karlsruhe.de/volltexte/1000012894.
[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), 330-336. doi: 10.1016/j.jmaa.2011.05.028.
[6]	W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003.
[7]	W.-P. Düll and G. Schneider, Justification of the Nonlinear Schrödinger equation for a resonant Boussinesq model, Indiana Univ. Math. J., 55 (2006), 1813-1834. doi: 10.1512/iumj.2006.55.2824.
[8]	J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities, Nonlinearity, 17 (2004), 551-565. doi: 10.1088/0951-7715/17/2/011.
[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), 493-523. doi: 10.3934/dcdsb.2006.6.493.
[10]	E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, Technical Report I., Los Alamos Rep, LA1940, Los Alamos, 1955; Reproduced in "Nonlinear Wave Motion" (ed. A. C. Newell), AMS, Providence, RI, 1974.
[11]	G. H. Golub and C. F. Van Loan, "Matrix Computations," Third edition, Johns Hopkins Studies in the Mathematical Sciences, John Hopkins University Press, Baltimore, MD, 1996.
[12]	T. Iguchi, A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves, Kyushu J. Math., 60 (2006), 267-303. doi: 10.2206/kyushujm.60.267.
[13]	A. Iserles, "A First Course in the Numerical Analysis of Differential Equations," Second edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.
[14]	G. James and P. Noble, Breathers on diatomic Fermi-Pasta-Ulam lattices, Physica D, 196 (2004), 124-171. doi: 10.1016/j.physd.2004.05.005.
[15]	L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium, Math. USSR Sbornik Surveys, 60 (1988), 457-483. doi: 10.1070/SM1988v060n02ABEH003181.
[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), 85-91. doi: 10.1017/S0308210500020989.
[17]	G. Schneider, Validity and limitation of the Newell-Whitehead equation, Math. Nachr., 176 (1995), 249-263. doi: 10.1002/mana.19951760118.
[18]	G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69-82. doi: 10.1007/s000300050034.
[19]	G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in "International Conference on Differential Equations," Vol. 1, 2 (Berlin, 1999), World Scientific Publ., River Edge, NJ, (2000), 390-404.
[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), 1475-1535. doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.
[21]	Guido Schneider and C. Eugene Wayne, The rigorous approximation of long-wavelength capillary-gravity waves, Arch. Ration. Mech. Anal., 162 (2002), 247-285. doi: 10.1007/s002050200190.
[22]	G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances, Journal of Differential Equations, 216 (2005), 354-386. doi: 10.1016/j.jde.2005.04.018.
[23]	G. Schneider, Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam-system, Applicable Analysis, 89 (2010), 1523-1539. doi: 10.1080/00036810903277150.
[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), 240-243. doi: 10.1103/PhysRevLett.15.240.

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