Painleve equations and integrable systems

Noumi和Yamada从仿射Weyl群对称性的角度系统地推广了Painleve型微分方程。这一结果发表在努米的书中，并激活了这一领域的研究。得到了第六类Painleve方程的一种新的Lax形式。在Painleve方程的仿射Weyl群的双态表示上发现了关于根系的普适结构。从高斯分解的角度解释了李氏理论的背景。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。这一构造证明了这种表示给出了由于Drinfeld-Sokolov族的相似约化而产生的Painleve型方程的对称性。另一方面，Kajiwara，Noumi，Yamada研究了具有Weyl群对称性的W(A^&lt；(1)&gt；_&lt；m-1&gt；×A^&lt；(1)&gt；_&lt；n-1&gt；)型Q-Painleve方程及其推广。这种表示是“热带”的(=自由减法)，并且通过超离散化有一些组合应用。得到了Q-Kp方程及其多项式解。Masuda给出了(Q-)Painleve V和VI方程的行列式公式。高野基于Backlund变换构造了初值空间。齐藤给出了初值空间的代数几何刻画。综上所述，对于项目的几乎所有问题，我们都得到了足够的结果。

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.