Journal d’Analyse Mathématique

, Volume 91, Issue 1, pp 105–121 | Cite as

An elementary and direct proof of the Painlevé property for the Painlevé equations I, II and IV

  • Jishan Hu
  • Min Yan
Article

Abstract

We present a direct and elementary proof that all the solutions of the Painlevé Equations I, II and IV are meromorphic functions on the whole complex plane. The proof uses some ideas from the existing proofs but applies the ideas in a different setting.

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References

  1. [1]
    A. Hinkkanen and I. Laine,Solutions of the first and second Painlevé equations are meromorphic, J. Analyse Math.79 (1999), 345–377.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Hinkkanen and I. Laine,Solutions of a modified third Painlevé equation are meromorphic, J. Analyse Math.85 (2001), 323–337.MATHMathSciNetGoogle Scholar
  3. [3]
    J. Hu and M. Yan,The mirror systems of integrable equations, Stud. Appl. Math.104 (2000), 67–90.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Hu and M. Yan,Singhularity analysis for integrable systems by their mirrors, Nonlinearity12 (1999), 1531–1543.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Painlevé,Mémoires sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. France28 (1900), 201–261.MathSciNetMATHGoogle Scholar
  6. [6]
    N. Steinmetz,On Painlevé’s equations I, II and IV, J. Analyse Math.82 (2000), 363–377.MATHMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2003

Authors and Affiliations

  • Jishan Hu
    • 1
  • Min Yan
    • 1
  1. 1.Department of MathematicsUniversity of Science and TechnologyHong Kong

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