Hirota’s Method and the Painlevé Property

  • J. D. Gibbon
  • M. Tabor
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 30)


Given a system of nonlinear ordinary or partial differential equations a most challenging problem is to find an analytical test to determine whether the given system is integrable. In the case of systems of o.d.e’s integrability (in the classical sense of “integration by quadratures” [1]) requires one to find as many integrals of the motion as the order of the system. However, in the case of Hamiltonian systems, owing to the special symplectic structure of phase space, a complete integration can be effected by the identification of as many involutive integrals as there are degrees of freedom (N). For integrable Hamiltonian systems the flow is confined to N-dimensional tori embedded in the 2N-dimensional phase space.


Integrable Hamiltonian System Schwarzian Derivative Toda Chain Inverse Scattering Transform Singular Manifold 
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  1. 1.
    E.T. Whittaker; Treatise on Analytical Dynamics, C.U.P., 1904.Google Scholar
  2. 2.
    V.E Zakharov and L. Faddeev; Funct. Anal. Appins. 5, 280 (1972).CrossRefGoogle Scholar
  3. 3.
    R. Hirota; in “Solitons”, eds: R.K. Bullough and P.J. Caudrey, Springer, Berlin (1980).Google Scholar
  4. 4.
    S. Kovalevskaya; Acta Mathematics, 12, 177, (1889), (Sur le problème de la rotation d’un corps solide autour d’un point fixe).Google Scholar
  5. 5.
    E.L. Ince; “Ordinary Differential Equations”, Dover, New York (1956).Google Scholar
  6. 6.
    M. Tabor and J. Weiss; Phys. Rev. A 24, 2157 (1981).CrossRefGoogle Scholar
  7. 7.
    Y.F. Chang, M. Tabor and J. Weiss; j:-Math. Phys., 23, 531 (1982).CrossRefGoogle Scholar
  8. 8.
    T. Bountis, H. Segur and F. Vivaldi; Phys. Rev., 257Ç 1257 (1982).Google Scholar
  9. 9.
    M.J. Ablowitz, A. Ramani and H. Segur; J. Math. Phys., 21, 715 (1980).CrossRefGoogle Scholar
  10. 10.
    J.B. McLeod and P.J. Olver; SIAM J. Math Anal., 14, 488, (1983).CrossRefGoogle Scholar
  11. 11.
    R. Nakach; Plasma Physics, Proc. 36th Nobel Symposium (ed. Wilhelmsson ), Plenum, N.Y., (1977).Google Scholar
  12. 12.
    J. Weiss, M. Tabor and G. Carnevale; J. Math. Phys., 24, 522, (1983).CrossRefGoogle Scholar
  13. 13.
    J. Weiss; J. Math. Phys., 24, 1405 (1983).CrossRefGoogle Scholar
  14. 14.
    J.D. Gibbon, P. Radmore, D. Wood and M. Tabor; to appear in Stud. in Appl. Math., (1984).Google Scholar
  15. 15.
    D.V. Chudnovsky, G.V. Chudnovsky and M. Tabor; Phys. Lett., 97A, 268 (1983).CrossRefGoogle Scholar
  16. 16.
    J. Weiss, J. Math. Phys., 25, 13 (1984).CrossRefGoogle Scholar
  17. 17.
    J.D. Gibbon and M. Tabor; preprint.Google Scholar
  18. 18.
    H. Wahlquist in “Bäcklund Transformations”; 515 R.M. Miura (Ed), Springer, Heidelberg (1976).Google Scholar
  19. 19.
    D.V. Chudnovsky and G.V. Chudnovsky; Proc. Nat. Acad. Sci., 80, 1774 (1983).CrossRefGoogle Scholar
  20. 20.
    M.M. Crum; Quart. J. Math. Oxford (2), 6, 121 (1955).CrossRefGoogle Scholar
  21. 21.
    M. Adler and J. Moser; Commun. Math. Phys. 41, 1 (1978).CrossRefGoogle Scholar
  22. 22.
    K. Okomoto; Physica 2D, 525 (1981).Google Scholar
  23. 23.
    A. Mikhailov; JETP Letts., 30, 414 (1979).Google Scholar
  24. 24.
    A.P. Fordy and J. Gibbons; Commun. Math. Phys., 77, 21 – 30 (1980)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • J. D. Gibbon
    • 1
  • M. Tabor
    • 2
  1. 1.Department of MathematicsImperial CollegeLondonGreat Britain
  2. 2.Department of Applied Physics & Nuclear EngineeringColumbia UniversityNew YorkUSA

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