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

The first, second and fourth Painleve 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 Painleve 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 Painleve equation, a well known Painleve'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. (C) 2015 Elsevier Inc. All rights reserved.