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Quasi-One-Dimensional Nonlinear Lattices

  • Leonid I. ManevitchEmail author
  • Agnessa Kovaleva
  • Valeri Smirnov
  • Yuli Starosvetsky
Chapter
Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)

Abstract

In this section, it is shown how the LPT concept can be extended to finite-dimensional oscillatory chains. The systems under consideration are finite-dimensional analogues of several classical infinite models which were initially used for analysis of such significant physical phenomena as recurrent energy transfer and localization.

References

  1. Ablowitz, M.J., Ladik, J.E.: Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17, 1011 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Akhmediev, N.N., Ankiewicz, A.: Solitosns: Nonlinear Pulses and Beams. Chapman and Hall, London (1992)Google Scholar
  3. Berman, G.P., Izrailev, F.M.: The Fermi–Pasta–Ulam problem: fifty years of progress. Chaos 15, 15104 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Berman, G.P., Kolovsky, A.R.: The limit of stochasticity for a one-dimensional chain of interacting oscillators. Sov. Phys. JETP 60, 1116 (1984)Google Scholar
  5. Bogolyubov, N.N., Mitropol’skii Yu.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon & Breach, Delhi (1961)Google Scholar
  6. Brillouin, L.: Wave Propagation in Periodic Structures. Dover Publications (1946)Google Scholar
  7. Budinsky, N., Bountis, T.: Stability of nonlinear modes and chaotic properties of 1D Fermi–Pasta–Ulam lattices. Physica D 8, 445 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Burlakov, V.M., Darmanyan, S.A., Pyrkov, V.N.: Modulation instability and recurrence phenomena in anharmonic lattices. Phys. Rev. B 54, 3257 (1996)CrossRefGoogle Scholar
  9. Chechin, G.M., Novikova, N.V., Abramenko, A.A.: Bushes of vibrational modes for Fermi–Pasta–Ulam chains. Physica D 166, 208 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Claude, Ch., Kivshar, YuS, Kluth, O., Spatschek, K.H.: Moving localized modes in nonlinear lattices. Phys. Rev. B 47, 14228 (1993)CrossRefzbMATHGoogle Scholar
  11. Dash, P.C., Patnaik, K.: Nonlinear wave in a diatomic Toda lattice. Phys. Rev. A, 23 (1981)Google Scholar
  12. Dauxois, T., Peyrard, M.: Physics of Solitons. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  13. Dauxois, T., Khomeriki, R., Piazza, F., Ruffo, S.: The anti-FPU problem. Chaos 15, 15110 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., Morris, H.S.: Solitons and Nonlinear Wave Equations. Academic Press Inc., London (1982)zbMATHGoogle Scholar
  15. Elliott, J.F.: The characteristic roots of certain real symmetric matrices. Mater thesis, University of Tennessee (1953)Google Scholar
  16. Feng, B.-F.: An integrable three particle system related to intrinsic.localized modes. J. Phys. Soc. Jpn. 75, 014401 (2006)CrossRefGoogle Scholar
  17. Fermi, E., Pasta, J., Ulam, S.: studies of the nonlinear problems. In: Segre, E. (ed.) Collected Papers of Enrico Fermi, p. 978. University of Chicago Press, Chicago (1965)Google Scholar
  18. Flach, S., Gorbach, A.V.: Discrete breathers—Advances in theory and applications. Phys. Rep. 467, 1 (2008)CrossRefzbMATHGoogle Scholar
  19. Flach, S., Willis, C.R.: Discrete breathers. Phys. Rep. 295, 181 (1998)MathSciNetCrossRefGoogle Scholar
  20. Flach, S., Ivanchenko, M.V., Kanakov, O.I.: q-Breathers and the Fermi-Pasta-Ulam problem. Phys. Rev. Lett. 95, 64102 (2005)CrossRefGoogle Scholar
  21. Gallavotti, G. (ed.): The Fermi-Pasta-Ulam Problem: A Status Report, vol. 728. Springer, Berlin (2008) (Springer Series Lect. Notes Phys.)Google Scholar
  22. Gregory, R.T., Karney, D.: A collection of matrices for testing computational algorithm. Wiley-Interscience (1969)Google Scholar
  23. Hajnal, D., Schilling, R.: Delocalization-localization transition due to anharmonicity. Phys. Rev. Lett. 101, 124101 (2008)CrossRefGoogle Scholar
  24. Henrici, A., Kappeler, T.: Results on normal forms for fpu chains. Commun. Math. Phys. 278, 145 (2008a)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Henrici, A., Kappeler, T.: Commun. Math. Phys. 278, 145 (2008b)CrossRefGoogle Scholar
  26. James, G., Kastner, M.: Bifurcations of discrete breathers in a diatomic Fermi–Pasta–Ulam chain. Nonlinearity 20, 631–657 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. James, G., Noble, P.: Breathers on di-atomic Fermi-Pasta-Ulam lattices. Physica D 196, 124–171 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Jensen, S.M.: The nonlinear coherent coupler. IEEE J Quantum Electron QE 18, 1580–1583 (1982)CrossRefGoogle Scholar
  29. Khusnutdinova, K.R.: Non-linear waves in a double row particle system. Vestnik MGU Math Mech 2, 71–76 (1992)zbMATHGoogle Scholar
  30. Khusnutdinova, K.R., Pelinovsky, D.E.: On the exchange of energy in coupled Klein-Gordon equations. Wave Motion 38, 1–10 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Kivshar, Y.S., Peyrard, M.: Modulational instabilities in discrete lattices. Phys. Rev. A 46, 3198 (1992)CrossRefGoogle Scholar
  32. Kosevitch, A.M., Kovalev, A.C.: Introduction to Nonlinear Physical Mechanics. Naukova Dumka, Kiev (1989). (in Russian)Google Scholar
  33. Kosevich, YuA, Manevich, L.I., Savin, A.V.: Wandering breathers and self-trapping in weakly coupled nonlinear chains: classical counterpart of macroscopic tunneling quantum dynamics. Phys. Rev. E 77, 046603 (2008)MathSciNetCrossRefGoogle Scholar
  34. Kovaleva, A., Manevitch, L.I., Manevitch E.L.: Intense energy transfer and superharmonic resonance in a system of two coupled oscillators. Phys. Rev. E 81(5) (2010)Google Scholar
  35. Kruskal, M.D., Zabusky, N.J.: Stroboscopic-perturbation procedure for treating a class of nonlinear wave equations. J. Math. Phys. 5, 231 (1964)MathSciNetCrossRefGoogle Scholar
  36. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 1. Pergamon, Oxford, UK (1960)Google Scholar
  37. Lichtenberg, A.J., Livi, R., Oettini, M., Ruffo, S.: Dynamics of oscillator chains. In: Gallavotti, G. (ed.) The Fermi-Pasta-Ulam Problem: A status report, vol. 728. Springer, Berlin (2008)Google Scholar
  38. Manevitch, L.I.: The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear Dyn. 25, 95 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Manevich, L.I.: In: Proceedings of the 8th Confence on Dynamical Systems: Theory and Applications, Lodz, 12–15 Dec 2005 (Poland, 2005), vol. 1, pp. 119–136 (2005)Google Scholar
  40. Manevitch, L.I.: In: Awrejcewicz, J., Olejnik, P. (eds.) 8th Conference on Dynamical Systems-Theory and Applications, DSTA-2005, p. 289 (2005)Google Scholar
  41. Manevitch, L.I.: New approach to beating phenomenon in coupled nonlinear oscillatory chains. Arch Appl Mech. 301 (2007)Google Scholar
  42. Manevitch, L.I.: Vibro-impact models for smooth non-linear systems. In: Ibrahim, R.A., Babitsky, V.I., Okuma, M. (eds.) Lecture Notes in Applied and Computational Mechanics, vol. 44, pp. 191–202. Springer, Berlin (2009)Google Scholar
  43. Manevitch, L.I., Gendelman, O.V.: Tractable Models of Solid Mechanics. Springer, Berlin (2011)Google Scholar
  44. Manevich, A.I., Manevich, L.I.: The Mechanics of Nonlinear Systems with Internal Resonances. Imperial College Press, London (2005) CrossRefzbMATHGoogle Scholar
  45. Manevitch, L.I., Musienko, A.I.: Limiting phase trajectories and energy exchange between an anharmonic oscillator and external force. Nonlinear Dyn. 58, 633–642 (2009)CrossRefzbMATHGoogle Scholar
  46. Manevitch, L.I., Oshmyan, V.G.: An asymptotic study of the linear vibrations of a stretched beam with concentrated masses and discrete elastic supports. J. Sound Vib. 223(5), 679–691 (1999)CrossRefGoogle Scholar
  47. Manevitch, L.I., Smirnov, V.V.: In: Indeitsev, D.A. (ed.) Advanced Problem in Mechanics, APM-2007, p. 289. IPME RAS, St.Petersburg (Repino), Russia (2007)Google Scholar
  48. Manevitch, L.I., Smirnov, V.V.: In: Awrejcewicz, J., Olejnik, P., Mrozowski, J. (eds.) 9th Conference on Dynamical Systems Theory and Applications, DSTA-2007, p. 301 (2007)Google Scholar
  49. Manevitch, L.I., Smirnov, V.V., 9th Conference on dynamical systems theory and applications. In: Awrejcewicz, J., Olejnik, P., Mrozowski, J. (eds.) 301, DSTA-2007, L’od’z, Poland 17–20 Dec 2007 (2007)Google Scholar
  50. Manevich, L.I., Smirnov, V.V.: Discrete breathers and intrinsic localized modes in small FPU systems. In: Proceedings of APM 2007: 293, St-Peterburg (2007)Google Scholar
  51. Manevitch, L.I., Smirnov, V.V.: Solitons in Macromolecular Systems. Nova Publ, New-York (2008)Google Scholar
  52. Manevitch, L.I., Smirnov, V.V.: Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains. Phys. Rev. E 82 (2010)Google Scholar
  53. Manevich, L.I., Mikhlin, Yu.V., Pilipchuk, V.V.: The Method of Normal Oscillations for Essentially Nonlinear Systems. Nauka, Moscow (1989) (in Russian)Google Scholar
  54. Manevitch, L.I., Mikhlin, YuV, Pilipchuk, V.N.: The Normal Vibrations Method For Essentially Nonlinear Systems. Nauka Publ, Moscow (1989). (in Russian)Google Scholar
  55. Manevitch, L.I., Kovaleva, A.S., Manevitch, E.L.: Limiting phase trajectories and resonance energy transfere in a system of two coupled oscillators. Math Problems Eng, vol. 2010. doi: 10.1155/2010/760479 (2010)
  56. Manevitch, L.I., Kovaleva, A.S., Shepelev, D.S.: Non-smooth approximations of the limiting phase trajectories for the duffing oscillator near 1:1 resonance. Accepted for publication in Physica D (2011)Google Scholar
  57. Mokrosst, F., Buttner, H.: Thermal conductivity in the diatomic toda lattice. J. Phys. C: Solid State Phys. 16, 4539–4546 (1983)CrossRefGoogle Scholar
  58. Nayfeh A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)Google Scholar
  59. Ovchinnikov, A.A., Erikhman, N.S., Pronin, K.A.: Vibrational-Rotational Excitations in Nonlinear Molecular Systems. Springer, Berlin (2001)CrossRefGoogle Scholar
  60. Poggi, P., Ruffo, S.: Exact solutions in the FPU oscillator chain. Physica D 103, 251 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  61. Rabinovich, M.I., Trubetskov, D.I.: Introduction to the Theory of Oscillations and Waves. Nauka, Moscow (1984). [in Russian]zbMATHGoogle Scholar
  62. Rink, B.: Symmetry and resonance in periodic FPU chains. Commun. Math. Phys. 218, 665 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  63. Rink, B., Verhulst, F.: Near-integrability of periodic FPU-chains. Phys. A 285, 467 (2000)CrossRefzbMATHGoogle Scholar
  64. Rosenberg, R.M.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Apl. Mech. 9, 156 (1966)Google Scholar
  65. Sandusky, K.W., Page, J.B.: Interrelation between the stability of extended normal modes and the existence of intrinsic localized modes in nonlinear lattices with realistic potentials. Phys. Rev. B 50, 866 (1994)CrossRefGoogle Scholar
  66. Scott A.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford University Press, Oxford (2003)Google Scholar
  67. Smirnov V.V., Manevitch L.I.: Limiting phase trajectories and dynamic transitions in nonlinear periodic systems, Acoust. Phys. 57, 271–276 (2011)Google Scholar
  68. Starosvetsky, Y., Manevitch L.I.: Nonstationary regimes in a Duffing oscillator subject to biharmonic forcing near a primary resonance. Phys. Rev. E 83 (2011)Google Scholar
  69. Starosvetsky, Y., Manevitch, L.I.: On intense energy exchange and localization in periodic FPU dimer chains, Physica D 264, 66–79 (2013)Google Scholar
  70. Toda, M.: Theory of Nonlinear Lattices, vol. 20. Springer, Berlin (1989)zbMATHGoogle Scholar
  71. Uzunov, I.M., Muschall, R., Gölles, M., Kivshar, Yuri S., Malomed, B.A., Lederer, F.: Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E 51, 2527–2537 (1995)CrossRefGoogle Scholar
  72. Vakakis, A.F.: Advanced Nonlinear Strategies for Vibration Mitigation and System Identification, 1st Ed. Springer, Berlin (2010)Google Scholar
  73. Vakakis, A.F., Manevitch, L.I., Mikhlin, YuV, Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Leonid I. Manevitch
    • 1
    Email author
  • Agnessa Kovaleva
    • 2
  • Valeri Smirnov
    • 1
  • Yuli Starosvetsky
    • 3
  1. 1.Institute of Chemical PhysicsRussian Academy of ScienceMoscowRussia
  2. 2.Space Research InstituteRussian Academy of ScienceMoscowRussia
  3. 3.Technion—Israel Institute of TechnologyFaculty of Mechanical EngineeringHaifaIsrael

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