第五类Painlevé方程解的渐近性态分析

Asymtotics of the general fifth Painleve equation

首先对Painleve方程求出数值解,然后用最小二乘法拟合出最佳渐近解,对最佳渐近解的表达式形式,用谐波平衡法方法得到振荡渐近解与参数之间的依赖关系.先前用此方法已对第三、四类Painleve方程的振荡渐近解做了一些研究.当参数α,β,δ,γ满足一些条件时,用同样的方法,对第五类Painleve方程给出了渐近解的形式,并找出这类渐近解与参数之间的关系.

The numerical solution for Painleve equation is given firstly, then the optimal asymptotics solution is found using the least square method. For expression of the optimal asymptotics solution, the relation of the oscillating solution depending on parameter is obtained by using the method of resonance wave equilibrium. By applying this method , the study was carried out for the scillating solutions of the third and four Painleve equations before. In this paper, when parameters α,β,δ,γ satisfy certain conditions, the oscillating solution for the fifth Painleve equation is found out by applying the same method in accordance with the relation of the parameter.