Painlevé Ping-Pong P3-P5

  • Ziemowit Popowicz
Part of the Lecture Notes in Physics book series (LNP, volume 189)


In this lecture I should like to discuss the concept of the Ping-Pong which appeared in the Painlevé equations. First let me explain that the name “Painlevé Ping-Pong”, in this context, is not mine but belongs to Prof. F. J. Bureau [1]. In the first part of my talk I shall give the explicit construction of this Ping-Pong for the third and fifth Painlevé equations. The Ping-Pong, fairly sketchily, is a special kind of the transformation and exists also for the other Painlevé equations [1,2]. It can be considered as the Bäcklund transformation for the third and fifth Painlevé equations [3]. In the second part I shall describe the possible physical applications of this Ping-Pong.


Ising Model Higgs Field Fourth Component Dimensional Ising Model Painlev6 Equation 
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  1. [1]
    F. J. Bureau, private communication.Google Scholar
  2. [2]
    M. J. Ablowitz and A. S. Fokas,On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23, 2033–2042 (1982).CrossRefGoogle Scholar
  3. [3]
    Z. Popowicz, The Bäcklund transformations for the third and fifth Painlevé equations, IFT University of Wroclaw preprint (1982).Google Scholar
  4. [4]
    N. Lukashevich, Differential Equations 3, 11 (1967); in Russian: 1913, (1967).Google Scholar
  5. [5]
    V. Gromak, Differential Equations 9, 2–373 (1975).Google Scholar
  6. [6]
    T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Spin-spin correlation functions for the twodimensional Ising model: exact theory in the scaling region. Phys. Rev. B13, 318–374 (1976).Google Scholar
  7. [7]
    H. B. Nielsen and P. Oleson, Vortex-line models for dual strings. Nucl. Phys. B61, 45–61 (1973).CrossRefGoogle Scholar
  8. [8]
    M. A. Lohe, Two-and three-dimensional instantons. Phys. Lett. 70B, 325–328 (1977).Google Scholar
  9. [9]
    C. Sacliĝlou, A string-like self-dual solution of Yang-Mills theory. Nucl. Phys. B178, 361–372 (1981).CrossRefGoogle Scholar
  10. [10]
    E. B. Bogomol'nyî, The stability of classical solutions. Soviet J. Nucl. Phys. 24, 449–454 (1976).Google Scholar
  11. [11]
    M. K. Prasad and C. M. Sommerfeld, Exact classical solution for the't Hooft monopole and the Julia-Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975).Google Scholar
  12. [12]
    C. N. Yang, Conditions of self-duality for SU(2) gauge fields on Euclidean four-dimensional space. Phys. Rev. Lett. 38, 1377–1379 (1977).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Ziemowit Popowicz
    • 1
  1. 1.Institute of Theoretical PhysicsUniversity of WroclawPoland

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