Reduction for Painlevé equations at the fixed singular points of the second kind
Reduction for Painlevé equations at the fixed singular points of the second kind
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第二类固定奇点处 Painlevé 方程的约简
DOI：
10.2969/jmsj/04230423
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发表时间：
1990
影响因子：
0.7
通讯作者：
K. Takano
中科院分区：
文献类型：
--
作者：
K. Takano
本文在Hamilton力学的框架下给出了Painleve方程在第二类不动点附近的一个简单约化定理。已知每个Painleve方程Pi（J=I，.，VI）等价于一个哈密顿系统：dA/dt=aH，/da，d~C/dt=-aH，/dA，其中哈密顿函数H =-H，（t，A，p）是A和p的多项式，其系数是t的有理函数（[14]）。我们称这些哈密顿系统为Painleve系统。然后对Painleve方程的固定奇点进行了形式化的分类：如果Painleve方程在该点的Poincare秩为零或正，则该点为第一类或第二类固定奇点.我们要构造每个Painleve系统在每个固定奇点处的2参数解族，换句话说，要获得一个局部双全纯变换，使其化为一个可解系统。众所周知，关于n-系统在固定奇点处的n-参数解族的构造，我们有J. Malmquist在所谓的Poincare条件下的一个一般定理（[12]，[8]）。然而，我们不能将定理应用于Painleve系统，因为它们完全违反了Poincare条件。近年来，受到M. Iwano（[9]），几位作者得到了Painleve系统在第二类不动点处的2-参数解族（[16]，[15]，[19]，[20]）。他们的作品，尤其是S.吉田从一般的角度解释了为什么Painleve系统的形式变换没有
This paper gives a simple reduction theorem for Painleve equations near the fixed singular points of the second kind in the framework of Hamiltonian mechanics. It is known that each Painleve equation P, (J=I, ... , VI) is equivalent to a Hamiltonian system : dA/dt=aH,/da, d~C/dt=-aH,/dA, where the Hamiltonian function H =-H,(t, A, p) is a polynomial of A and p of which the coefficients are rational functions of t ([14]). We call these Hamiltonian systems Painleve systems. Then the fixed singular points are formally classified as follows : a fixed singular point of a Painleve equation is o f the first kind or o f the second kind if Poincare rank of the corresponding Painleve system at the point is zero or positive respectively. We want to construct a 2-parameter family of solutions of each Painleve system at each fixed singular point, in other words, to obtain a local biholomorphic transformation which reduces it to a solvable system. As is well known, concerning the construction of an n-parameter family of solutions of an n-system at a fixed singular point, we have a general theorem by J. Malmquist under the so-called Poincare's condition ([12], [8]). However, we can not apply the theorem to Painleve systems because Poincare's condition is completely violated for them. Recently, having been stimulated by the idea of M. Iwano ([9]), several authors have obtained 2-parameter families of solutions of Painleve systems at the fixed singular points of the second kind ([16], [15], [19], [20]). Their works especially those by S. Yoshida explain, from a general point of view, the reason why the formal transformations for Painleve systems without
潘勒夫方程世界
DOI：
--
发表时间：
2020
期刊：
影响因子：
--
作者：
照井章;小原功任;田島慎一;坂井秀隆;坂井秀隆
通讯作者：
坂井秀隆