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

Abstract

The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ 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 $\C P^3(p,q,r,s)$. 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.

References in corpus (2)

• Kovalevskaya Exponents and Poisson Structures
• Painleve Test and the Resolution of Singularities for Integrable Equations

Cited by in corpus (4)

• The Third, Fifth and Sixth Painlevé Equations on Weighted Projective Spaces
• Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation
• On an Orbifold Hamiltonian Structure for the First Painleve Equation
• Weights, Kovalevskaya exponents and the Painlevé property