Birational canonical transformations and classical solutions of the sixth Painlevé equation

• Humihiko Watanabe ^[1]
  1. [1] Kyushu University

     Kyushu University

     Hakata Ku, Japón

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 27, Nº 3-4, 1998, págs. 379-425
• Idioma: inglés
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• Resumen

  • Two topics on the sixth Painleve equation are treated in this paper. In Section 1, a simple construction of a group of birational canonical transformations of the sixth equation isomorphic to the affine Weyl group of D4 root system is given by exploiting an affine Weyl group symmetry of the Hamiltonian structure of the sixth equation defined on the defining variety of the equation. In the rest of this paper (Sections 2-4), based on Umemura’s theory on algebraic differential equations, all one-parameter families of classical solutions of the sixth equation are determined, and the irreducibility of the sixth equation is proved. The latter is a rigorous proof of what Painleve asserted in C. R. Acad. Sci. Paris 143 (1906), 1111-1117.