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Preface
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 |
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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