2013/11/08 by Hayato Chiba, Chiba, Hayato · 4 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Differential Equations and Numerical Methods #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #math.CA #math.DG
paper · pdf · doi:10.48550/arxiv.1311.1877
openalex publication_date 2013/11/08 · arxiv created 2014/07/05 · arxiv updated 2014/07/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces \C P3(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 \C P3(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.