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Zbl 0597.34004
Murata, Yoshihiro
Rational solutions of the second and the fourth Painlevé equations. (English)
[J] Funkc. Ekvacioj, Ser. Int. 28, 1-32 (1985). ISSN 0532-8721

This paper establishes by group theory methods the conditions under which the second and fourth Painlevé equations $$ P\sb 2(\alpha):\frac{d\sp 2\lambda}{dt\sp 2}=2\lambda\sp 3+t\lambda +\alpha$$ $$ P(\alpha,\theta):\frac{d\sp 2\lambda}{dt\sp 2}=\frac{1}{2\lambda}(\frac{d\lambda}{dt})\sp 2+\frac\quad {3}{2}\lambda\sp 3+4t\lambda\sp 2+2(t\sp 2-\alpha)\lambda - \frac{2\theta\sp 2}{\lambda} $$ have rational solutions for $\lambda$ (t). Specifically $P\sb 2(\alpha)$ has a unique rational solution if and only is $\alpha$ is an integer and $P\sb 4(\alpha,\theta)$ likewise for specifically characterised values of $\alpha$ and $\theta$. In each case explicit solutions are given for small values of $\alpha$ and $\theta$.
[G.G.Wake]

MSC 2000:: *34M55 Painlevé and other special equations
34A34 Nonlinear ODE and systems, general
34A05 Methods of solution of ODE

Keywords: second Painlevé equation; group theory methods; fourth Painlevé equations; rational solutions

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