Global Analysis of Painleve Equations

1.Painleve方程的对称性：建立了Painleve方程的Backlund变换(仿射Weyl群的实现)理论。该理论不仅为Painleve方程的研究提供了很好的视角，而且提供了有用的工具。例如，我们可以很容易地得到每个Backlund变换的形式作为双曲变换以及与Painleve方程相关的各种特殊多项式。类似的理论现在正在为离散的Painleve方程而发展。初始条件空间：(1)阐明了初始条件空间与Backlund变换的关系，即用Backlund变换修补仿射图得到的流形与Okamoto的初始条件空间同构。由此，我们可以得到在仿射Weyl群下参数等价的初始条件空间是同构的。(2)得到了高阶A^&lt；(1)&gt；_4型Painleve方程和二元退化Garnier系统的初始条件空间。精确WKB分析：(1)用精确WKB分析解决了具有大参数的第二类Painleve方程的连接问题。将第一类大参数Painleve方程的联结公式合成，给出了联结公式。为此，证明了Birkhoff范式的一个约化定理。将常用的最速下降法推广到三阶线性微分方程解。超几何方程：提出了在所有参数都是纯虚数的情况下研究Schwarz理论的问题。并进行了实验研究。

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.