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

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• 作者：Hayato Chiba ^{chiba@imi.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：The Painlev&eacute ; equations ; Weighted projective space
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：260
• 期：2
• 页码：1263-1313
• 全文大小：643 K

文摘

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)$\mathbb{C}{P}^3(p,q,r,s)$ with suitable weights (p,q,r,s)$(p,q,r,s)$ 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)$\mathbb{C}{P}^3(p,q,r,s)$. 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.