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

Hayato Chiba

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6 Citations (Scopus)

Abstract

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
Volume260
Issue number2
DOIs
Publication statusPublished - Jan 15 2016

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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