Annali di Matematica Pura ed Applicata

, Volume 146, Issue 1, pp 337–381 | Cite as

Studies on the Painlevé equations

I.-Sixth Painlevé equation PVI
  • Kazuo Okamoto


In this series of papers, we study birational canonical transformations of the Painlevé system ℋ, that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also τ -function related to ℋ and particular solutions of ℋ. The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of ℋ associated with the affine transformations of the space of parameters of ℋ, we obtain the non-linear representation: G→G*, of the affine Weyl group of the exceptional root system of the type F4 A canonical transformation of G* can extend to the correspondence of the τ -functions related to ℋ. We show the certain sequence of τ -functions satisfies the equation of the Toda lattice. Solutions of ℋ, which can be written by the use of the hypergeometric functions, are studied in details.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1985

Authors and Affiliations

  • Kazuo Okamoto
    • 1
  1. 1.Department of Mathematics, College of Arts and SciencesUniversity of TokyoTokyoJapan

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