2018 Fiscal Year Final Research Report
Structure of asymptotic solutions of integrable systems and WKB analysis
| Project/Area Number |
26287015
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Partial Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Doshisha University (2017-2018)
Kyoto University (2014-2016) |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
神本 晋吾 京都大学, 数理解析研究所, 研究員 (10636260)
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| Research Collaborator |
KAMIMOTO Shingo
AOKI Takashi
KOIKE Tatsuya
KAWAI Takahiro
HIROSE Sampei
JOSHI Nalini
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| Project Period (FY) |
2014-04-01 – 2019-03-31
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| Keywords | 解析学 / 関数方程式論 / 漸近解析 / 代数解析 / 可積分系 / ホロノミック系 / パンルヴェ方程式 / ＷＫＢ解析 |
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| Outline of Final Research Achievements |
To extend the exact WKB analysis to systems of partial differential equations including nonlinear equations and difference equations, we study several integrable systems and hypergeometric systems from the viewpoint of the exact WKB analysis. Consequently we obtain deeper understanding for coalescing phenomena of turning points for nonlinear equations and, furthermore, the following new results are also obtained: discovery of the appearance of non-hereditary double turning points associated with the restriction of holonomic systems, determination of the explicit form of connection formulas for Stokes phenomena of discrete Painleve equations, and a new idea about the analytic interpretation of instanton-type formal solutions of Painleve equations in terms of elliptic functions.
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| Free Research Field |
数物系科学
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| Academic Significance and Societal Importance of the Research Achievements |
線型常微分方程式に比して、非線型方程式や差分方程式、さらに多変数の偏微分方程式系に対する漸近解析はまだまだ発展途上である。本研究で得られた種々の成果は、いずれもこうしたより一般の微分差分方程式系への完全ＷＫＢ解析の拡張に向けて大きな一歩となるものと考えられる。特に、楕円函数を利用したパンルベ方程式のインスタントン型形式解の解析的意味付けに関するアイデアは、非線型方程式に対する漸近解析を革新する可能性を秘めた重要な成果である。
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