Geometric study of Painleve equations and infinite integrable systems

2016 Fiscal Year Final Research Report

• Project/Area Number: 25870234
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis; Mathematical analysis
• Research Institution: Hitotsubashi University
• Principal Investigator: Tsuda Teruhisa 一橋大学, 大学院経済学研究科, 准教授 (00452730)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: パンルヴェ方程式 / 超幾何函数 / 無限可積分系 / 有理函数近似 / 連分数

Outline of Final Research Achievements

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve; equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

Free Research Field

数物系科学