| Project/Area Number |
11440047
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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 |
Basic analysis
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| Research Institution | Faculty of Science, Kobe Univ. |
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Principal Investigator |
YAMADA Yasuhiko Faculty of Science, Professor, 理学部, 教授 (00202383)
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| Co-Investigator(Kenkyū-buntansha) |
TAKANO Kyouichi Faculty of Science, Professor, 理学部, 教授 (10011678)
SAITO Masahiko Faculty of Science, Professor, 理学部, 教授 (80183044)
NOUMI Masatoshi Graduate School of Science and Technology, Professor, 自然科学研究科, 教授 (80164672)
KAJIWARA Kenji Kyushu Univ. Graduate School of Mathematics, associate prof., 数理学研究院, 助教授 (40268115)
MASUDA Tetsu Graduate School of Science and Technology, assistant prof., 自然科学研究科, 助手 (00335457)
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| Project Period (FY) |
1999 – 2002
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| Keywords | Painleve equation / integrable systems / discrete Painleve equation / Backlund transformation / Weyl group / birational representation / ultra-discretization / space of initial value |
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| Research Abstract |
Noumi and Yamada gave a systematic generalization of Painleve-type differential equations from the point of view of affine Weyl group symmetry. This result is presented in the Noumi's book and activate the research of this area. A new Lax formalism for the sixth Painleve equation is also obtained. The universal structure with respect to the root systems was discovered on the birational representation of the affine Weyl group arising from Painleve equations. Lie theoretic background is also explained based on the gauss decomposition. The representation was lifted to the tau-functions. The tau functions are certain matrix elements of affine Lie algebras. This construction proved that the representation gives the symmetry of the Painleve type equations arising as the similarity reduction of the Drinfeld-Sokolov hierarchy. On the other hand, Kajiwara, Noumi, Yamada studied the q-Painleve IV equation and its generalization with Weyl group symmetry of type W (A^<(1)>_<m-1> × A^<(1)>_<n-1>). This representation is "tropical" (=subtraction free) and has some combinatorial applications through the ultra-discretization. q-KP hierarchy and their polynomial solutions are obtained. Masuda gave the determinant formulas for the (q-)Painleve V and VI equations. Takano constructed the space of initial value based on the Backlund transformations. Saito gave the algebro-geometric characterization of the space of initial value. In summary, we obtained sufficient results for almost all the problems of the project.
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