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# Rational Solutions of Painlevé Equations

Published:2002-06-01
Printed: Jun 2002
• Wenjun Yuan
• Yezhou Li
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## Abstract

Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$, $\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation~(P$_6$) in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and B\"acklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlev\'e equations (P$_1$)--(P$_5$) below, which have been investigated by many authors [1]--[7].
 Keywords: Painlevé differential equation, rational function, Bäcklund transformation
 MSC Classifications: 30D35 - Distribution of values, Nevanlinna theory 34A20 - unknown classification 34A20

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