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.

诺米（Noumi）和山田（Yamada）从仿射Weyl群对称的角度进行了系统的泛型型微分方程的系统概括。该结果在Noumi的书中提出，并激活该领域的研究。还获得了第六次帕克斯方程的新的宽松格式。关于根系的普遍结构是在阵亡系统的十亿个代表性上发现的。还根据高斯分解来解释谎言理论背景。表示表示为tau功能。 tau函数是仿射为代数的某些矩阵元素。这种构造规定，该表示形式给出了疼痛型方程的对称性，作为Drinfeld-Sokolov层次结构的相似性降低。另一方面，Yamada的Kajiwara，Noumi，Noumi研究了Q-Painleve IV方程及其用Weyl oft type W（a^<（1）> _ <m-1>×A^<（1）> _ <n-1>）的Weyl组对称性的概括。该表示形式是“热带”（=免除），并且通过超散制具有一些组合应用。获得Q-KP层次结构及其多项式溶液。 Masuda给出了（Q-）Painleve V和VI方程的确定公式。 Takano基于反向插件变换构建了初始值的空间。斋藤给出了初始值空间的代数几何表征。总之，我们几乎为项目的所有问题获得了足够的结果。