Integrable Systems and WKB Analysis

在本项目中，在京都大学教授T. Kawai （RIMS, Kyoto university）和Nishikawa先生（Kyoto university研究生）的帮助下，我们主要研究了精确WKB分析对可积系统高阶Painleve方程的推广。首先，对于由最简并的Garnier系统得到的（P_I）层次、由KdV层次化简得到的（P_<II>）层次和Noumi-Yamada系统（即（P_<IV>）和（P_V）层次）等高阶painlevel方程的几个层次，我们发现这些非线性方程的拐点和Stokes曲线（“Stokes几何”）与相关Lax对的拐点和Stokes曲线密切相关。其次，发现高阶Painleve方程的Stokes曲线存在交叉，并在交叉点处产生一条新的Stokes曲线；新的Stokes曲线可以理解为从虚拐点发出的Stokes曲线，它也可以用相关Lax对的Stokes几何来表征。最后，我们证明了（P_I）和（P_<II>）层次中任意成员的0参数解（即不含任何自由参数的代数构造形式解）在第一类简单拐点附近可以局部转化为传统（P_I）方程的解。检验这些结果是否适用于作为Lax对相容条件的更一般的非线性方程将是今后的主要问题。在上述研究的同时，我们还研究了以下相关课题：(1)精确最陡下降法的改进，线性方程新Stokes曲线的检测方法；(2)线性方程系统的精确WKB分析及其在转移概率计算中的应用；(3)WKB型微微分方程的精确WKB分析。

In this project, with the aid of Prof. T. Kawai (RIMS, Kyoto Univ.) and Mr. Y. Nishikawa (a graduate student of Kyoto Univ.), we have mainly studied generalization of exact WKB analysis to higher order Painleve equations that are obtained from integrable systems. First, for several hierarchies of higher order Painleve equations such as the (P_I) hierarchy obtained from the most degenerate Garnier system, the (P_<II>) hierarchy obtained by reduction of the KdV hierarchy, and the Noumi-Yamada system (i.e., (P_<IV>) and (P_V) hierarchy), we have found that turning points and Stokes curves ("Stokes geometry") of these nonlinear equations are closely related with those of the associated Lax pair. Secondly, it is discovered that Stokes curves of higher order Painleve equations do cross and a new Stokes curve emanates from a crossing point. A new Stokes curve can be understood as a Stokes curve emanating from a virtual turning point, which is also characterized in terms of the Stokes geometry of the associated Lax pair. Lastly, we have shown that a 0-parameter solution (i.e., an algebraically constructed formal solution without any free parameter) of any member of (P_I) and (P_<II>) hierarchies can be locally transformed to that of the traditional (P_I) equation near a simple turning point of the first kind. To examine if these results hold for more general nonlinear equations arising as compatibility condition of Lax pairs will be a main problem in future. In parallel with the above researches, we have also studied the following related subjects: (i) refinement of the exact steepest descent method, a method detecting new Stokes curves of linear equations, (ii) exact WKB analysis for systems of linear equations and its application to computations of transition probabilities, and (iii) exact WKB analysis for microdifferential equations of WKB type.