Geometric Riemann scheme of the Painlevé equations

Details Geometric Riemann scheme of the Painlevé equations Geometric Riemann scheme of the Painlevé equations

2007-09-05

arxiv.org/abs/0709.0597v7

Mathematics

General Mathematics

29 pages, 3 figures

Scientific paper

In this paper, we introduce the notion of geometric Riemann scheme of the sixth Painlev\'e equation, which consists of the pair of accessible singular points and matrix of linear approximation around each singular point on the boundary divisor in the Hirzebruch surface. Giving this in the differential system satisfying certain conditions, we can recover the Painlev\'e VI system with the polynomial Hamiltonian. We give a generalization of the Painlev\'e VI system by generalizing the geometric Riemann scheme of the sixth Painlev\'e equation. This system has movable branch points. Nevertheless, we show that this system has rich birational symmetries. We also consider the case of the Painlev\'e V,IV and III systems. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular points. We show the existence theorem of these equations, which can be considered as a non-linear version of the existence theorem of Fuchsian differential equations from the viewpoint of geometrical property.

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