Bäcklund Transformations and the Painlevé Property

  • John Weiss
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 25)


For systems with the Painlevé Property, Bäcklund transformations can be defined. These appear as specializations (truncations) of certain expansions of the solution about its’ singular manifold. With reference to the Lax pair for a system, the Bácklund transformations are equivalent to transformations of linear systems developed by Darboux (Bäcklund-Darboux transformations).

For specific systems the Bäcklund-Darboux transformations lead to a reformulation of these systems in terms of the Schwarzian derivative. We find the Bäcklund transformations of these system and study their periodic fixed points.

The periodic fixed points of the Bäcklund transformations determine a finite dimensional invariant manifold for the flow of the system. The resulting (ordinary) differential equations have a hamiltonian structure and the flow of the (partial) differential system is represented by commuting flows on the finite dimensional manifold.


Toda Lattice Schwarzian Derivative Singular Manifold Backlund Transformation Miura Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Kowalevseaya, Acta Mathematica, 14 (1890), pp. 81.MathSciNetCrossRefGoogle Scholar
  2. [2]
    E.L. Incb, Ordinary Differential Equations, Dover, New York, 1956.Google Scholar
  3. [3]
    M.J Ablowitz, A. Ramani and H. Segur, J. Math. Phys., 21 (1981), pp. 715.MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J.B. Mcleod and P.J. Olver, Siam J. Math. Anal., 14 (1983), pp. 1566.MathSciNetCrossRefGoogle Scholar
  5. [5]
    W.F. Osgood, Topics in the Theory of Functions of Several Complex Variables, Dover, New York, 1966.Google Scholar
  6. [6]
    J. Weiss, M. Tabor and G. Carnevale, J. Math. Phys., 24 (1983), pp. 522.MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. [7]
    E. Aille, Ordinary Differential Equations in the Complex Plane, John Wiley, New York, 1976.Google Scholar
  8. [8]
    J. Weiss, J. Math. Phys., 24 (1983), pp. 1405.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. [9]
    J. Weiss, J. Math. Phys., 25 (1984), pp. 13.MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. [10]
    J. Weiss, J. Math. Phys., 26 (1985), pp. 258.MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. [11]
    J. Weiss, J. Math. Phys., 25 (1984), pp. 2226.MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. [12]
    R.S. Ward, Phys Lett., 102A (1984), pp. 279.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R.S. Ward, Phil. Trans. R. Soc. Lond. A, 315 (1985), pp. 451.ADSzbMATHCrossRefGoogle Scholar
  14. [14]
    R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, 1962.zbMATHGoogle Scholar
  15. [15]
    E.V. Doktorov and S. Yu. SAEovICH, J. Phys. A, 18 (1985), pp. 3327.ADSzbMATHCrossRefGoogle Scholar
  16. [16]
    P.A. Clarkson, Phys. Lett., 109A (1985), pp. 205.CrossRefGoogle Scholar
  17. [17]
    P.A. Clarkson, Physica, 18D (1986), pp. 209.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Willy Hereman, private communication.Google Scholar
  19. [19]
    M. Lavie, Canadian J. Math., 21 (1969), pp. 235.zbMATHGoogle Scholar
  20. [20]
    G. Darboux, Théorie Générale des Surfaces, II, Chelsea, New York, 1972.Google Scholar
  21. [21]
    P. Deift and E. Trubowitz, Comm. Pure Appl. Math., 32 (1979), pp. 121.MathSciNetzbMATHGoogle Scholar
  22. [22]
    J. Weiss, Phys. Lett., 102A (1984), pp. 329.MathSciNetCrossRefGoogle Scholar
  23. [23]
    J. Weiss, Phys. Lett., 105A (1984), pp. 387.MathSciNetCrossRefGoogle Scholar
  24. [24]
    J. Weiss, J. Math. Phys., 27 (1986), pp. 1293.MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. [25]
    M.J. Ablowitz and H. Segue, Solitons and the Inverse Scattering Transformation, SIAM, Philadelphia, 1981.Google Scholar
  26. [26]
    B. Gaffet, Physica, 26D (1987), pp. 123.MathSciNetzbMATHGoogle Scholar
  27. [27]
    B. Gaffet, J. Phys. A, 21 (1988), pp. 2491.MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. [28]
    A.C. Newell, M. Tabor and Y.B. Zeng, Physica, 29D (1987), pp. 1.MathSciNetzbMATHGoogle Scholar
  29. [29]
    J.D. Gibbon, A.C. Newell, M. Tabor and Y.B. Zeng, Nonlinearity, 1 (1988), pp. 481.MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [30]
    R. Conte, Phys. Letts, 184A (1988), p. 100.MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    George Wilson, Phys. Letts, 132A (1988), pp. 445.ADSzbMATHCrossRefGoogle Scholar
  32. [32]
    J. Weiss, J. Math. Phys., 27 (1986), pp. 2647.MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. [33]
    P. Davis, Circulant Matricies, Wiley, New York, 1983.Google Scholar
  34. [34]
    J. Weiss, J. Math. Phys., 28 (1987), pp. 2025.MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. [35]
    J. Weiss, Bäcklund transformations, Focal surfaces and the Two dimensional Toda lattice, submitted to, Phys Letts. A (1988).Google Scholar
  36. [36]
    J. Weiss, WORK IN PROGRESS.Google Scholar
  37. [37]
    A.P. Fordy and J. Gibbons, Commun. Math. Phys., 77 (1980), pp. 21.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • John Weiss
    • 1
  1. 1.USA

Personalised recommendations