2000 Fiscal Year Final Research Report Summary
Global Analysis of Painleve Equations
| Project/Area Number |
10440058
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kobe University |
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Principal Investigator |
TAKANO Kyoichi Kobe Univ Faculty of Science, Professor, 理学部, 教授 (10011678)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Yasuhiko Kobe Univ. Grad. School of S.& T., Asso. Prof, 自然科学研究科, 助教授 (00202383)
NOUMI Masatoshi Kobe Univ. Grad. School of S.& T., Professor, 自然科学研究科, 教授 (80164672)
SASAKI Takeshi Kobe Univ. Faculty of Science, Professor, 理学部, 教授 (00022682)
TAKEI Yoshitsugu Kyoto Univ. RIMS, Asso. Prof., 数理解析研究所, 助教授 (00212019)
IWASAKI Katsunori Kyushu Univ. Dept. of Math., Professor, 数理学研究科, 教授 (00176538)
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| Project Period (FY) |
1998 – 2000
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| Keywords | Painleve equations / Higher order Painleve equations / Garnier systems / Affine Weyl group symmetries / Backlund transformations / Spaces of initial conditions / Hamiltonian structures / Exact WKB analysis |
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| Research Abstract |
1. Symmetries of Painleve equations : Theory of Backlund transformations (realization of affine Weyl groups) for Painleve equations has been constructed. The theory gives not only good perspective but also usefull tools to the study of Painleve equations. For example, we can easily obtain the form of each Backlund transformation as birational transformation and various kinds of special polynomials associated with Painleve equations. Similar theory is now being devepoled for discrete Painleve equations.
2. The spaces of initial conditions : (1) A relation between the spaces of initial conditions and Backlund transformations has been made clear, namely, the manifold obtained by patching affine charts via Backlund transformations are isomorphic to Okamoto's space of initial conditions. Fromx this fact, we can derive that the spaces of initial conditions whose papameters are equivalent under the affine Weyl group are isomorphic to each other. (2) Spaces of initial conditions for a higher order Painleve equation of type A^<(1)>_4 and degenerated Garnier systems of two variables have been obtained.
3. Exact WKB analysisi : (1) The connection problem for the second Painleve equation with a large paraneter has been solved by the use of exact WKB analysis. The connection formulas are given by compositions of those for the first Painleve equation with a large parameter. For this purpose, a reduction theorem to Birkhoff's normal form has been shown. The usual steepest descent method has been extended to one for third order linear differential equations.
4. Hypergeometric equations : A problem of studying Schwarz theory in the case where all parameters are pure imaginary numbers has been proposed. Some experiments were carried out.
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