Global Analysis of Painleve Equations

1. Painleve方程的对称性：Painleve方程的Backlund变换（仿射Weyl群的实现）理论已经建立。该理论为Painleve方程的研究提供了良好的前景和有用的工具。例如，我们可以很容易地得到每一个Backlund变换的形式作为双有理变换和各种特殊的多项式与Painleve方程。类似的理论现在正在开发离散Painleve方程。初始条件空间：（1）明确了初始条件空间与Backlund变换之间的关系，即用Backlund变换修补仿射图得到的流形与Okamoto初始条件空间同构。从这一事实出发，我们可以推导出仿射外尔群下参数相等的初始条件空间彼此同构。(2)得到了高阶A^<（1）>_4型Painleve方程和退化的二元Garnier方程组的初始条件空间.精确的WKB分析：（1）用精确WKB分析方法解决了第二类大参数Painleve方程的连接问题。利用第一类大参数Painleve方程的连接公式的合成，给出了连接公式。为此目的，一个减少定理Birkhoff的规范形式已被证明。将通常的最速下降法推广到三阶线性微分方程.超几何方程：提出了在所有参数都是纯虚数的情况下研究施瓦茨理论的问题。进行了一些实验。

1. Symmetries of Painleve equations : Theory of Backlund transformations (realization of affine Weyl groups) for Painleve equations has been constructed. The theory gives not only good perspective but also usefull tools to the study of Painleve equations. For example, we can easily obtain the form of each Backlund transformation as birational transformation and various kinds of special polynomials associated with Painleve equations. Similar theory is now being devepoled for discrete Painleve equations.2. The spaces of initial conditions : (1) A relation between the spaces of initial conditions and Backlund transformations has been made clear, namely, the manifold obtained by patching affine charts via Backlund transformations are isomorphic to Okamoto's space of initial conditions. Fromx this fact, we can derive that the spaces of initial conditions whose papameters are equivalent under the affine Weyl group are isomorphic to each other. (2) Spaces of initial conditions for a higher order Painleve equation of type A^<(1)>_4 and degenerated Garnier systems of two variables have been obtained.3. Exact WKB analysisi : (1) The connection problem for the second Painleve equation with a large paraneter has been solved by the use of exact WKB analysis. The connection formulas are given by compositions of those for the first Painleve equation with a large parameter. For this purpose, a reduction theorem to Birkhoff's normal form has been shown. The usual steepest descent method has been extended to one for third order linear differential equations.4. Hypergeometric equations : A problem of studying Schwarz theory in the case where all parameters are pure imaginary numbers has been proposed. Some experiments were carried out.