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Painleve equations and integrable systems

Painleve 方程和可积系统

基本信息

批准号：
11440047
负责人：
YAMADA Yasuhiko
金额：
$ 7.87万
依托单位：
Faculty of Science, Kobe Univ.
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

项目摘要

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.

Noumi和Yamada从仿射Weyl群对称性的角度对Painleve型微分方程进行了系统推广。这一结果在Noumi的书中得到呈现，并激活了该领域的研究。还获得了第六 Painleve 方程的新 Lax 形式。关于根系统的通用结构是在 Painleve 方程产生的仿射 Weyl 群的双有理表示上发现的。李理论背景也基于高斯分解进行了解释。该表示被提升为 tau 函数。 tau 函数是仿射李代数的某些矩阵元素。这种构造证明了该表示给出了由于 Drinfeld-Sokolov 层次结构的相似性约简而产生的 Painleve 型方程的对称性。另一方面，Kajiwara、Noumi、Yamada 研究了 q-Painleve IV 方程及其与 W 型 Weyl 群对称性的推广 (A^<(1)>_<m-1> × A^<(1)>_<n-1>)。这种表示是“热带的”（=无减法），并且通过超离散化具有一些组合应用。获得q-KP层次及其多项式解。 Masuda 给出了 (q-)Painleve V 和 VI 方程的行列式。高野基于贝克兰德变换构建了初值空间。 Saito给出了初值空间的代数几何表征。综上所述，我们对项目的几乎所有问题都获得了足够的结果。