Exact WKB analysis for higher order Painleve equations

为了建立准确的WKB分析painlevel层次结构，我们研究了1。一个高阶Painleve方程的Stokes几何及其潜在的Lax对，2。高阶painlevel方程自由参数形式解的构造；在（简单）拐点附近的结构解，从而得到如下结果。首先，对于1。得到了第一个painlevel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Lax对的大小大于2的Noumi-Yamada系统，其Lax对的虚拐点也与非线性方程Stokes几何的确定有关，如S.Sasaki所指出的。与本田的联合研究表明，虚拟拐点和Lax对的新Stokes曲线的作用可以通过引入“树结构”等图理论概念来很好地理解。第二，关于2。，我们成功地构造了瞬态解，将二阶情况下的方法，即利用哈密顿系统到其Birkhoff范式的方法推广到高阶方程。通过这种方法，我们得到了具有自由参数的高阶painlevel方程的形式解，这些方程可表示为像第一个painlevel层次结构那样的hamilton系统。最后，至于3。将Lax对大小为2的painlevel层次0参数解的第一类简单拐点结构定理推广到Noumi-Yamada系统。将结构定理推广到瞬态解和第二类拐点分析是未来的重要问题；如果克服了这些问题，高阶painlevel方程的连接问题将得到明确的解决。

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.