TY - JOUR

T1 - The first, second and fourth Painlevé equations on weighted projective spaces

AU - Chiba, Hayato

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2016/1/15

Y1 - 2016/1/15

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jde.2015.09.020

DO - 10.1016/j.jde.2015.09.020

M3 - Article

AN - SCOPUS:84947868161

SN - 0022-0396

VL - 260

SP - 1263

EP - 1313

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -