Exact WKB analysis for higher order Painleve equations

为了建立精确的WKB分析Painleve层次结构，我们研究了1。高阶Painleve方程及其Lax对的Stokes几何，2.构造了高阶Painleve方程带自由参数的形式解，3.研究了（简单）转向点附近解的结构，得到了如下结果：首先，对于1.，我们得到了第一个Painleve层次（其Lax对具有最简单的结构）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对的虚拟转向点也与非线性方程的Stokes几何的确定有关，如S. Sasaki所指出的。与N.本田的联合工作现在正在阐明，通过引入“树结构”等图论概念，可以很好地理解Lax对的虚拟转折点和新Stokes曲线的作用。我们通过扩展在二阶情况下采用的方法成功地构造了瞬子型解，即，即把哈密顿系统简化为Birkhoff标准形，再简化为高阶方程。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，这些方程可以用第一Painleve族等Hamilton系统的形式表示。将Lax对大小为2的Painleve方程族的0-参数解在第一类简单转向点处的结构定理推广到Noumi-Yamada方程组.将该结构定理推广到瞬子型解以及第二类转向点处的分析是未来的重要课题.如果克服了这些问题，高阶Painleve方程的连接问题将得到显式解决.

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.