### Abstract

The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces CP^{3}(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 CP^{3}(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 language | English |
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Pages (from-to) | 1263-1313 |

Number of pages | 51 |

Journal | Journal of Differential Equations |

Volume | 260 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 15 2016 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Differential Equations*,

*260*(2), 1263-1313. https://doi.org/10.1016/j.jde.2015.09.020