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Kajiwara, Kenji; Noumi, Masatoshi; Yamada, Yasuhiko

Geometric aspects of Painlevé equations. (English) Zbl 1441.34095

J. Phys. A, Math. Theor. 50, No. 7, Article ID 073001, 164 p. (2017).
The paper is a survey of the geometric aspects of the Painlevé equations, with emphasis on the discrete Painlevé equations. The authors remind the geometric approach to Painlevé equations of Okamoto and Sakai, the Painlevé property, the isomonodromic deformations etc. The theory of discrete Painlevé equations relies on the geometry of the initial values space which is an eight point configuration in \(\mathbb{P}^1\times \mathbb{P}^1\) classified according to the degeneration of points. The authors explain the roles played in this theory by the affine Weyl group symmetries, of hypergeometric solutions and Lax pairs, of Picard lattices and root systems, of Bäcklund transformations and \(\tau\) functions. They provide a collection of basic data: equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.
Reviewer: Vladimir P. Kostov (Nice)

Cited in 1 Review
Cited in 79 Documents

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
14H70 Relationships between algebraic curves and integrable systems
33C20 Generalized hypergeometric series, \({}_pF_q\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
39A10 Additive difference equations
39A13 Difference equations, scaling (\(q\)-differences)

Keywords:

Painlevé equations; discrete Painlevé equations; space of initial values; root systems; Lax pair; hypergeometric functions; affine Weyl groups
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References:

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