第三类Painlevé方程解的渐近性态分析

Asymtotics analysis of the general third Painlevé equation

通过数值方法,结合理论分析,给出了第三类Painlev啨方程y″=y′2y-y′x+1x(αy2+β)+γy3+δy.振荡渐近解的表达形式:当δ>0,γ<0时,y=A+B｜x｜-1/2cos(a｜x｜+bln｜x｜+c)+O(｜x｜-1,x→±∞;当δ=0,γ<0时,y=｜x｜-1/3[A+B｜x｜-1/3cos(a｜x｜2/3+bln｜x｜+c)]+O(x-1),x→±∞;当δ>0,γ=0时,y=｜x｜1/3[A+B｜x｜-1/3cos(a｜x｜2/3+bln｜x｜+c)]+O(｜x｜-1/3),x→±∞.

The expression of the asymptotical vibrational solution of the third Painlevé equation y″=y′~2y-y′x+1x(αy^2+β)+γy^3+δy is given as follows: if δ>0,γ<0,y=A+B |x|^(-1/2)cos(a|x|+bln|x|+c)+O(|x|^(-1)),x→±∞;if δ=0,γ<0,y=|x|^(-1/3)\[A+B|x|^(-1/3)cos(a|x|^(2/3)+bln|x|+c)\]+O(x^(-1)),x→±∞;if δ>0,γ=0,y=|x|^(1/3)\[A+B|x|^(-1/3)cos(a|x|^(2/3)+bln|x|+c)\]+O(|x|^(-1/3)),x→±∞. 