2006 Fiscal Year Final Research Report Summary
Mathematical theory of dynamical systems of Painleve type
| Project/Area Number |
15340057
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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 | KYUSHU UNIVERCITY |
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Principal Investigator |
KAJIWARA Kenji Kyushu University, Faculty of Mathematics, Associate Professor, 大学院数理学研究院, 助教授 (40268115)
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| Co-Investigator(Kenkyū-buntansha) |
NOUMI Masatoshi Kobe University, Department of Mathematics, Professor, 大学院自然科学研究科, 教授 (80164672)
YAMADA Yasuhiko Kobe University, Department of Mathematics, Professor, 理学部, 教授 (00202383)
IWASAKI Katsunori Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (00176538)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Painleve equations / discrete Painleve equations / τ functions / hypergeometric functions / elliptic curve / affine Weyl group / integrable systems / chaotic systems |
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| Research Abstract |
1. We have extended the theory of symmetric form for q-Painleve IV equation with (A_2+A_1)^<(1)> affine Weyl group symmetry to formulate the q-KP hierarchy. By the similarity reduction we have constructed the hierarchy of discrete systems with (A_<m-1>+A_1)^<(1)> affine Weyl group symmetry, and further, that with (A_<m-1>+A_<n-1>)^<(1)> affine Weyl group symmetry.
2. We have presented a formulation of the elliptic Painleve equation and its generalizations, the former of which is located on the top of all the Painleve systems of second order. Namely, we have formulated the time evolution and the Backlund transformations as Cremona transformations on the configuration space of generic points in the complex projective space, and given their realization as birational transformations parametrized by the theta functions on the level of the τ functions. We have also formulated the time evolution as the addition formula on the moving pencil of plane cubic curves, and clarified the geometric mea
… More
ning of the Hamilitonians of the Painleve differential equations.
3. Applying the above formulation, we have constructed the simplest hypergeometric solutions to the elliptic Painleve equation and all the q-Painleve equations, and completed the list of coalscent diagram of hypergeometric functions, starting from the elliptic hypergeometric function _<10>E_9 to the q-Airy function.
4. Combining the algebro-geometric formulation of the Painleve VI equation and the ergodic theory of birational mapping on the algebraic surface by using the Riemann-Hilbert correspondence, we have shown that the nonlinear monodromy of the Painleve VI equation is chaotic along almost all the loops.
5. We have shown that the entries of Hankel determinant formula for the solutions of the Painleve differential equations arise as coefficients of asymptotic expansion of the ratio of solutions to the auxiliary linear problem, and that this phenomenon originates from the structure of the KP hierarchy.
6. Applying the above results we have discussed some new extentions or new solutions to the discrete and ultradiscrete Toda equation. Less
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Research Products
(52 results)
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[Book] 共形場理論入門2005
Author(s)
山田 泰彦
Total Pages
265
Publisher
培風館
Description
「研究成果報告書概要(和文)」より
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