Painleve equations and integrable systems

Noumi和Yamada从仿射Weyl群对称的角度对painleve型微分方程进行了系统的推广。这一结果在野美的书中被提出，并激活了这一领域的研究。得到了第六阶Painleve方程的一个新的Lax形式。在Painleve方程的仿射Weyl群的双元表示上发现了根系的普遍结构。在高斯分解的基础上，说明了李理论的背景。表示法被提升为tau函数。函数是仿射李代数的矩阵元素。该构造证明了该表示给出了由Drinfeld-Sokolov层次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W型(A^<（1)>_<m-1> × A^<(1)>_<n-1>）的Weyl群对称q-Painleve IV方程及其推广。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。得到了q-KP层次及其多项式解。Masuda给出了（q-） painlev和VI方程的行列式公式。Takano基于Backlund变换构造了初值空间。Saito给出了初值空间的代数-几何表征。总之，我们对项目的几乎所有问题都得到了充分的结果。

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.