The first, second and fourth Painlevé equations on weighted projective spaces

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9 Citations (Scopus)


The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces CP3(p,q,r,s) with suitable weights (p, q, r, s) determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of the equations, a simple proof of the Painlevé property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of CP3(p,q,r,s). In particular, for the first Painlevé equation, a well known Painlevé's transformation is geometrically derived, which proves to be the Darboux coordinates of a certain algebraic surface with a holomorphic symplectic form. The affine Weyl group, Dynkin diagram and the Boutroux coordinates are also studied from a view point of the weighted projective space.

Original languageEnglish
Pages (from-to)1263-1313
Number of pages51
JournalJournal of Differential Equations
Issue number2
Publication statusPublished - 2016 Jan 15
Externally publishedYes


  • The Painlevé equations
  • Weighted projective space

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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