Exact WKB analysis for higher order Painleve equations

为了建立精确的Painleve层次的WKB分析，我们研究了1。高阶Painleve方程及其Lax对的Stokes几何，2.构造高阶Painleve方程的带自由参数的形式解，3.在(简单)转折点附近的解的结构，从而得到以下结果.首先，对于1，我们得到了第一个Painleve族(其Lax对具有最简单结构)的Stokes几何的完整描述.另一方面，对于Noumi-Yamada系统(其Lax对的大小大于2)，Lax对的虚拟转折点也与确定非线性方程的Stokes几何有关，正如S.Sasaki所指出的那样。与N.Honda的合作现在澄清了Lax对的虚拟转折点和新的Stokes曲线的作用可以通过引入图论的概念如树结构来很好地理解。第二，对于2，我们成功地通过将二阶情况下所使用的方法，即利用哈密顿系统的约化为其Birkhoff范式，来构造瞬子类型的解。通过这种方法，我们得到了高阶Painleve方程的带自由参数的形式解，这些形式解可以用类似第一Painleve方程的哈密顿系统的形式来表示。最后，对于3，将Lax对为2的Painleve方程的0参数解的第一类简单转折点的结构定理推广到Noumi-Yamada系统。将结构定理推广到瞬时子型解和在第二类转折点的分析是未来的重要问题；如果它们被克服，高阶Painleve方程的联系问题将被显式地解决。

To establish exact WKB analysis for Painleve hierarchies, we studied1. the Stokes geometry of a higher order Painleve equation and its underlying Lax pair,2. construction of formal solutions with free parameters to a higher order Painleve equation,3. the structure of solutions near (simple) turning points,and consequently obtained the following results.Firstly, as for 1., we obtained a complete description of the Stokes geometry for the first Painleve hierarchy (whose Lax pair has the simplest structure). On the other hand, for Noumi-Yamada systems (whose Lax pair is of size greater than two) virtual turning points of the Lax pair are also relevant to the determination of the Stokes geometry of nonlinear equations, as is pointed out by S.Sasaki. The joint work with N.Honda is now clarifying that the roles of virtual turning points and new Stokes curves of the Lax pair can be well understood by introducing graph-theoretical notions such as "tree structure".Secondly, as for 2., we succeeded in constructing instanton-type solutions by extending the method employed in the second order case, i.e., that of using reduction of a Hamiltonian system to its Birkhoff normal form, to higher order equations. Through this method we obtained formal solutions with free parameters for higher order Painleve equations that are expressible in the form of Hamiltonian systems like the first Painleve hierarchy.Finally, as for 3., the structure theorem at a simple turning point of the first kind for 0-parameter solutions of Painleve hierarchies whose Lax pair is of size two was generalized to Noumi-Yamada systems.Generalization of the structure theorem to instanton-type solutions and analysis at turning points of the second kind are important future problems; if they are overcome, connection problems for higher order Painleve equations will be solved explicitly.