Geometric aspects of Painlevé equations

Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada

Research output: Contribution to journalReview articlepeer-review

63 Citations (Scopus)


In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev 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 degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric 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.

Original languageEnglish
Article number073001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number7
Publication statusPublished - Jan 12 2017

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)


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