Singular-perturbative reduction to Birkhoff normal form and instanton-type formal solutions of Hamiltonian systems
Singular-perturbative reduction to Birkhoff normal form and instanton-type formal solutions of Hamiltonian systems
复制标题
奇异微扰约简到哈密顿系统的 Birkhoff 范式和瞬子型形式解
DOI：
10.2977/prims/1195144426
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发表时间：
1998
影响因子：
1.2
通讯作者：
Yoshitsugu Takei
中科院分区：
文献类型：
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作者：
Yoshitsugu Takei
如通过二阶Fuchsian方程的单值群的计算所说明的（参见图1）。[AKT 1]），精确WKB分析为研究线性常微分方程解的整体性态提供了一个强有力的工具。为了将这种分析推广到非线性方程，T。Kawai（RIMS，京都大学），T.青木（近畿大学）在我们的系列文章（[KT 1]，[AKT 2]，[KT 2]）中，作者发展了大参数Painleve方程的WKB理论。(See[T1]，[T2]也是。）在我们的处理中，通过[AKT 2]中的多尺度分析构造的称为瞬子型解的2参数形式解起着核心作用。虽然在文献[KT 2]中我们已经成功地分析了它们在简单转向点附近的局部结构，但它们的一些重要性质，如在固定奇点附近的行为还没有得到阐明。为了研究它们在固定正则型奇点附近的行为，本文提出了一种新的构造大参数Painleve方程的2-参数形式解的方法.我们在这里提出的形式解的新构造是基于高野的工作[Tkal]（也见[Tka 2]），在那里他构造了一个2参数族的解析解在每个正则型奇点（普通）Painleve方程。他利用Painleve方程可以写成Hamilton系统的形式这一众所周知的事实（我们在这里称之为Painleve Hamilton系统），并建立了一些约化定理，
As is illustrated by the computation of monodromy groups of second-order Fuchsian equations (cf. [AKT1]), the exact WKB analysis provides us with a powerful tool for studying global behavior of solutions of linear ordinary differential equations. To generalize such an analysis to nonlinear equations, T. Kawai (RIMS, Kyoto Univ.), T. Aoki (Kinki Univ.) and the author have developed the WKB theory for Painleve equations with a large parameter in our series of articles ([KT1], [AKT2], [KT2]). (See [Tl], [T2] also.) In our treatment 2-parameter formal solutions called instanton-type solutions, which were constructed through the multiple-scale analysis in [AKT2], are playing a central role. Although we have succeeded in analyzing their local structure near simple turning points in [KT2], some of their important properties such as the behavior near fixed singular points have not been clarified yet. In this paper, to investigate their behavior near fixed regular-type singular points, we propose a new construction of 2-parameter formal solutions of Painleve equations with a large parameter. The new construction of formal solutions we propose here is based on Takano's work [Tkal] (see [Tka2] also), where he constructed a 2-parameter family of analytic solutions at each regular-type singular point of (ordinary) Painleve equations. He made use of the well-known fact that Painleve equations can be written in the form of Hamiltonian systems (which we call Painleve Hamiltonian systems here) and established some reduction theorem for