2018 Fiscal Year Final Research Report
Research of integrable systems around the Painleve equations
| Project/Area Number |
15K04894
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
SAKAI Hidetaka 東京大学, 大学院数理科学研究科, 准教授 (50323465)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | パンルヴェ方程式 / 差分方程式 / 特殊函数 / 超幾何函数 |
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| Outline of Final Research Achievements |
In this research project we published two papers in the journals.
The first one is a joint research with M. Jimbo and H. Nagoya, which is a research that has constructed a general solution of a q-difference Painleve equation.In the case of differential equations, the solutions of deformation equation systems of isomonodromic deformation are constructed using 5-point conformal blocks, and solutions are obtained from their expansions. We have considered a similar theory for the q-difference case.
The second one is a collaborative research with M. Yamaguchi, which is a study on the classification theory of linear q-difference equations. In this paper, the definition of the spectral type to guide the classification of linear equations, and the definition of the transformation of the equation using Jackson's integral called the middle convolution, are given. We also showed that this transformation has appropriate properties.
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| Free Research Field |
特殊函数論
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| Academic Significance and Societal Importance of the Research Achievements |
非線型の函数方程式については，線型の同様な問題と比べて有効な一般論を構築することが難しい．解となる函数の具体的な性質にいたっては，代数函数や超幾何函数などのよく知られた特殊函数によって具体的に記述できる特別な場合を除くと，なかなか解析ができないのが現状である．q差分パンルヴェ方程式の一般解が構成できたことは，具体的な計算に向けての重要な手がかりを与えることになる．
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