2015/09/30 by Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada · 2 citations
Mathematics · Physics and Astronomy · #math-ph #math.CA #math.MP #msc:14H70 #msc:33C20 #msc:33D15 #msc:34M55 #msc:37K20 #msc:39A10 #msc:39A13 #nlin.SI
paper · pdf · doi:10.1088/1751-8121/50/7/073001
published as J. Phys. A: Math. Theor. 50(7) (2017) 073001 · 168 pages. Some errors found in the published version are corrected
arxiv created 2017/01/22 · arxiv updated 2017/01/24
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on ℙ1×ℙ1 and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.