Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 627–633 | Cite as

Two-Wave Interactions in the Fermi–Pasta–Ulam Model

  • S. D. Glyzin
  • S. A. Kashchenko
  • A. O. Tolbey


This work is devoted to investigating the dynamic properties of the solutions to the boundaryvalue problems associated with the classic Fermi–Pasta–Ulam (FPU) system. We analyze these problems for an infinite-dimensional case where a countable number of roots of characteristic equations tend to an imaginary axis. Under these conditions, we build a special nonlinear partial differential equation that acts as a quasi-normal form, i.e., determines the dynamics of the original boundary-value problem with the initial conditions in a sufficiently small neighborhood of the equilibrium state. The modified Korteweg–deVries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation act as quasi-normal forms depending on the parameter values. Under some additional assumptions, we apply the renormalization procedure to the boundary-value problems obtained. This procedure leads to an infinite-dimensional system of ordinary differential equations. We describe a method for folding this system into a special boundary- value problem, which is an analog of the normal form. The main contribution of this work is investigating the interaction of the waves moving in different directions in the FPU problem by using analytical methods of nonlinear dynamics. It is shown that the mutual influence of the waves is asymptotically small, does not affect their shape, and contributes only to a shift in their speeds, which does not change over time.


Fermi–Pasta–Ulam model generalized KdV equation quasi-normal form boundary-value problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Russel, S.J., Report of waves, Report 14th Meeting of the British Association for the Advancement of Science, London, 1844, pp. 311–390Google Scholar
  2. 2.
    Fermi, E., Pasta, J.R., and Ulam, S., Studies of Nonlinear Problems, Report LA-1940, Alamos Scientific Laboratory, 1955.CrossRefMATHGoogle Scholar
  3. 3.
    Porter, M.A., Zabusky, N.J., Hu, B., and Campbell, D.K., Fermi, Pasta, Ulam and the birth of experimental mathematics, Am. Sci., 2009, vol. 97, no. 3, pp. 214–221.CrossRefGoogle Scholar
  4. 4.
    Dauxois, T., Peyrard, M., and Ruffo, S., The Fermi–Pasta–Ulam “numerical experiment:” History and pedagogical perspectives, 2005. arXiv: nlin/0501053v2.Google Scholar
  5. 5.
    Genta, T., Giorgilli, A., Paleari, S., and Penati, T., Packets of resonant modes in the Fermi–Pasta–Ulam system, Phys. Lett. A, 2012, vol. 376, pp. 2038–2044.CrossRefMATHGoogle Scholar
  6. 6.
    Kudryashov, N.A., Fermi–Pasta–Ulam model and higher-order nonlinear evolution equations, Vestn. Nats. Issled. Yad. Univ. Mosk. Inzh.-Fiz. Inst., 2016, vol. 5, no. 1, pp. 3–22.MathSciNetGoogle Scholar
  7. 7.
    Kudryashov, N.A., Analytical Theory of Nonlinear Differential Equations, Moscow-Izhevsk: Institute of Computer Science, 2004.Google Scholar
  8. 8.
    Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M., Phys. Rev. Lett., 1967, vol. 19, pp. 1095–1097.CrossRefGoogle Scholar
  9. 9.
    Ablowitz, M.J. and Clarkson, P.A., Solitons Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.CrossRefMATHGoogle Scholar
  10. 10.
    Kudryashov, N.A., Refinement of the Korteweg–de Vries equation from the Fermi–Pasta–Ulam model, Phys. Lett. A, 2015, vol. 279, pp. 2610–2614.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kudryashov, N.A., From the Fermi–Pasta–Ulam model to higher-order nonlinear evolution equations, Rep. Math. Phys., 2016, vol. 77, no. 1, pp. 57–67.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Boca Raton, FL: Chapman and Hall/CRC, 2011.MATHGoogle Scholar
  13. 13.
    Volkov, A.K. and Kudryashov, N.A., Nonlinear waves described by a fifth-order equation derived from the Fermi–Pasta–Ulam system, Comput. Math. Math. Phys., 2016, vol. 56, no. 4, pp. 680–687.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kudryashov, N.A., Ryabov, P.N., and Sinelshchikov, D.I., Nonlinear waves in media with fifth order dispersion, Phys. Lett. A, 2011, vol. 375, pp. 2051–2055.CrossRefMATHGoogle Scholar
  15. 15.
    Kashchenko, S.A., Normal form for the KdV–Burgers equation, Dokl. Math., 2016, vol. 93, no. 3, p. 331.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kashchenko, S.A., On the quasi-normal forms for parabolic equations with small diffusion, Rep. Acad. Sci. USSR, 1988, vol. 299, pp. 1049–1053.Google Scholar
  17. 17.
    Kaschenko, S.A., Normalization in the systems with small diffusion, Int. J. Bifurcations Chaos, 1996, vol. 6, no. 7, pp. 1093–1109.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kashchenko, I.S. and Kashchenko, S.A., Quasi-normal forms of two-component singularly perturbed systems, Dokl. Math., 2012, vol. 86, no. 3, p. 865.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kashchenko, I.S., Multistability in nonlinear parabolic systems with low diffusion, Dokl. Math., 2010, vol. 82, no. 3, p. 878.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ablowitz, M.J. and Segur, H., Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1981.CrossRefMATHGoogle Scholar
  21. 21.
    Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., and Morris, H.C., Solitons and Nonlinear Wave Equations, London: Academic Press, 1982.MATHGoogle Scholar
  22. 22.
    Newell, A.C., Solitons in Mathematics and Physics, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1985.CrossRefMATHGoogle Scholar
  23. 23.
    Zabusky, N.J. and Kruskal, M.D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys Rev. Lett., 1965, vol. 15, pp. 240–243.CrossRefMATHGoogle Scholar
  24. 24.
    Kudryashov, N.A., Methods of Nonlinear Mathematical Physics, Dolgoprudnyi: Izd. dom Intellekt, 2010.Google Scholar
  25. 25.
    Korteweg, D.J. and de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new tipe of long stationary waves, Phil. Mag., 1895, vol. 39, pp. 422–443.CrossRefMATHGoogle Scholar
  26. 26.
    Burgers, J.M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1948, vol. 1, pp. 171–199.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rabinovich, R.S. and Trubetskov, D.I., Introduction in the Theory of Oscillations and Waves, Izhevsk: RCD, 2000.MATHGoogle Scholar
  28. 28.
    Kudryashov, N.A., On “new travelling wave solutions” of the KdV and the KdV–Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 2009, vol. 14, no. 5, pp. 1891–1900.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kudryashov, N.A., Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl. Math. Mech., 2009, vol. 52, no. 3, pp. 361–365.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kudryashov, N.A., One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer., 2012, vol. 17, pp. 2248–2253.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kudryashov, N.A., Painleve analysis and exact solutions of the Korteweg–de Vries equation with a source, Appl. Math. Lett., 2015, vol. 41, pp. 41–45.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., Autowave processes in continual chains of unidirectionally coupled oscillators, Proc. Steklov Inst. Math., 2014, vol. 285, pp. 81–98.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., Buffering effect in continuous chains of unidirectionally coupled generators, Theor. Math. Phys., 2014, vol. 181, no. 2, pp. 1349–1366.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  • S. D. Glyzin
    • 1
    • 2
  • S. A. Kashchenko
    • 1
    • 3
  • A. O. Tolbey
    • 1
  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia
  2. 2.Scientific Center in ChernogolovkaRussian Academy of SciencesChernogolovka, Moscow oblastRussia
  3. 3.National Research Nuclear University MEPhIMoscowRussia

Personalised recommendations