Painleve equations and integrable systems

Noumi和Yamada从仿射Weyl群对称性的角度给出了Painleve型微分方程的系统推广。这一结果被发表在Noumi的书中，并激活了这一领域的研究。第六Painleve方程的一个新的Lax形式也得到了。根系统的普遍结构是在由Painleve方程产生的仿射Weyl群的双有理表示上发现的。基于高斯分解，说明了其理论背景。该表示被提升到τ函数。τ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作为Drinfeld-Sokolov层次的相似性减少而产生的Painleve型方程的对称性。另一方面，Kajiwara，Noumi，Yamada研究了q-Painleve IV方程及其推广，具有W（A^&lt;（1）&gt;_<m-1>× A^&lt;（1）&gt;_<n-1>）型Weyl群对称性。这种表示是“热带”（=减法自由），并通过超离散化有一些组合应用。得到了q-KP族及其多项式解。增田给出了（q-）Painleve V和VI方程的行列式公式。Takano基于Backlund变换构造了初值空间。齐藤给出了空间的初始值的代数几何特征。总之，我们得到了足够的结果，几乎所有的问题的项目。

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.