Some Aspects of Soliton Dynamics

Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 30)


It is about 30 years since the recent study of nonlinear waves started with the work by FERMI et al. [1]. They numerically integrated the equations of motion of certain one-dimensional nonlinear lattices, and found recurrence to initial state at least for small nonlinearity and smooth initial excitation. The idea that for sufficiently smooth waves, a lattice can be approximated by a continuum led ZABUSKY and KRUSKAL [2] to the numerical study of the Korteweg-de Vries equation
$$\partial u/\partial t + u\partial u/\partial x + {\delta ^2}{\partial ^3}u/\partial {x^3} = 0.$$


Soliton Solution Nonlinear Evolution Equation Solitary Wave Solution Cnoidal Wave Infinite Lattice 
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  1. 1.
    E. Fermi, J. Pasta and S. Ulam, Collected Papers of E. Fermi, Univ. of Chicago Press, 1965Google Scholar
  2. 2.
    N. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 240 (1965)CrossRefGoogle Scholar
  3. 3.
    C.S. Gardner, J.M. Greene, M.D. Kruskal and R. Miura, Phys. Rev. Lett. 19 1095 (1967)CrossRefGoogle Scholar
  4. 4.
    M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Studies Appl. Math. 53 249 (1974)Google Scholar
  5. 5.
    M. Toda, Theory of Nonlinear Lattices, Springer Series of Solid-State Sciences vol. 20 (1981)Google Scholar
  6. 6.
    H. Flaschka, Phys. Rev. B9 1924 (1974)Google Scholar
  7. 7.
    E. Date and S. Tanaka, Prog. Theor. Phys. 55 457 (1976); Prog. Theor. Phys. Suppl. 59 107 (1976)CrossRefGoogle Scholar
  8. 8.
    M. Kac and P. van Moerbeke, Proc. Nat. Acad. Sci. USA 72 1627, 2879 (1975)CrossRefGoogle Scholar
  9. 9.
    N. Saitoh, J. Phys. Soc. Japan 49 409 (1980)CrossRefGoogle Scholar
  10. 10.
    A. Ramani, B. Dorizzi and B. Grammaticos, Phys. Rev. Lett. 49 1539 (1982)CrossRefGoogle Scholar
  11. 11.
    M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 71 313 (1981)CrossRefGoogle Scholar
  12. 12.
    H. Yoshida, Celes. Math. 31 363, 381 (1983), Nonlinear Integrable Systems (ed. by M. Jimbo and T. Miwa) p.273, World Scientific 1983Google Scholar
  13. 13.
    R. Hirota and J. Satsuma, Prog. Theor. Phys. Suppl. 59 64 (1976)CrossRefGoogle Scholar
  14. 14.
    see for example, E. Date, M. Kashiwara, M. Jimbo and T. Miwa in Nonlinear Integrable Systems loc. cit., p.39Google Scholar
  15. 15.
    T. Taniuti and C.C. Wei, J. Phys. Soc. Japan 24 941 (1968)CrossRefGoogle Scholar
  16. 16.
    H. Ono, J. Phys. Soc. Japan 39 1082 (1975); J.Satsuma in Nonlinear Integrable Systems loc. cit., p.183Google Scholar
  17. 17.
    M. Toda, R. Hirota and J. Satsuma, Prog. Theor. Phys. Suppl. 59 148 (1976)CrossRefGoogle Scholar
  18. 18.
    S. Watanabe and M. Toda, J. Phys. Soc. Japan 50 3443 (1981)CrossRefGoogle Scholar
  19. 19.
    M. Toda and N. Saitoh, J. Phys. Soc. Japan 52 3703 (1983)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • M. Toda
    • 1
  1. 1.Shibuya-ku, Tokyo 151Japan

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