Exact WKB analysis for higher order Painleve equations

高阶 Painleve 方程的精确 WKB 分析

基本信息

批准号：
16540148
负责人：
TAKEI Yoshitsugu
金额：
$ 2.24万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540148/
关键词：
Painleve hierarchy Exact WKB analysis Noumi-Yamada system Stokes geometry Virtual turning point Instanton-type solution Birkhoff normal form Structure theorem 高階Painleve方程式 Lax Pair Stokes図形変換論第1Painleve方程式

项目摘要

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.

为了建立精确的WKB分析，用于帕恩莱夫层次结构，我们研究了1。高阶painleve方程及其基础宽松对的stokes几何形状，2。具有自由参数的正式解决方案的构建，以达到更高阶段的Painleve方程，3。解决方案靠近（简单）转弯点的结构，因此获得了以下结果。首先，我们获得了第一个Painleve层次结构的Stokes几何形状的完整描述（其LAX对具有最简单的结构）。另一方面，对于Noumi-Yamada系统（其宽松对大于两个），LAX对的虚拟转弯点也与S.Sasaki指出的非线性方程的Stokes几何形状的确定也有关。与N. Honda的联合合作现在澄清，可以通过介绍图形理论概念（例如“树结构”）来很好地理解lax对的虚拟转折点和新的Stokes曲线的作用。第二，我们成功地通过在二阶情况下构建了二阶案例，即在section case，即，i.e.e.e.e.e.e.e.e of second case，即，i.e.e.e.e.e.e.e.e ex case，is of the sy.e.e.e.e.e.e of。形式，高阶方程。通过这种方法，我们获得了具有自由参数的正式解决方案，用于高阶次要方程式，这些方程式以哈密顿系统的形式表达，例如第一个painleve层次结构。第二类转折点的激体型解决方案和分析是重要的未来问题；如果克服了它们，则将明确解决高阶Painleve方程的连接问题。

项目成果

期刊论文数量（53）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Toward the exact WKB analysis for higher-order Painleve equations - The case of Noumi-Yamada systems

高阶 Painleve 方程的精确 WKB 分析 - Noumi-Yamada 系统的案例

DOI：
发表时间：
2004
期刊：
Publications of the Research Institute for Mathematical Sciences, Kyoto University 40巻
影响因子：
0
作者：
I.Miyamoto;M.Yanagishita;Y.Takei
通讯作者：
Y.Takei

WKB analysis of higher order Painleve equations with a large parameter

大参数高阶 Painleve 方程的 WKB 分析

DOI：
发表时间：
2006
期刊：
Advances in Mathematics 203巻
影响因子：
0
作者：
Hirano;N;Shioji;N;Y. Takei;Y. Takei;Y. Takei;Y. Takei;Y. Takei;Y. Takei and H. Wakako;Y.Takei;Y. Takei;T. Kawai,and Y. Takei
通讯作者：
T. Kawai,and Y. Takei

On a local reduction of a higher order Painleve equation and its under-lying Lax pair near a simple turning point of the first kind

关于高阶 Painleve 方程及其底层 Lax 对在第一类简单转折点附近的局部约简