Research of integrable systems around the Painleve equations

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03323
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: The University of Tokyo
• Principal Investigator: Sakai Hidetaka 東京大学, 大学院数理科学研究科, 准教授 (50323465)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: パンルヴェ方程式 / 差分方程式 / 特殊函数 / 超幾何函数

Outline of Final Research Achievements

In this research project, one paper has been published in an academic journal and another paper is in preparation.
In the published paper, in collaboration with T. Mase and A. Nakamura, we proposed a discrete Hamiltonian for the discrete Painleve equation and showed that the equation can be easily written by using it.
The paper being prepared is a joint research with T. Hosoi. We determined the form of a fourth-order homogeneous quadratic differential equation (under some simple assumption) that has only t = 0, 1, infinity as singular points, all of which are of type (H). It contains the bilinear form of the 6th Painleve equation.

Free Research Field

特殊函数論

Academic Significance and Societal Importance of the Research Achievements

パンルヴェ6型方程式は非常に複雑な形をした方程式であり，その具体形を何らかの特徴づけから求めるのには大変な計算を要することが多い．細井氏との共同研究の結果は，特異点の近くにおける方程式の局所的な様子だけから方程式を決定できるという意味で，フックス型線型微分方程式における超幾何微分方程式の特徴づけを想起させ，面白い事象だと思う．