Tohoku Mathematical Journal
2008
December
SECOND SERIES VOL. 60, NO. 4

Tohoku Math. J.
60 (2008), 581-595

Title NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND PAINLEVÉ TYPE WITH THE QUASI-PAINLEVÉ PROPERTY

Author Shun Shimomura

(Received March 3, 2008, revised June 4, 2008)
Abstract. We present a class of nonlinear differential equations of second Painlevé type. These equations, with a single exception, admit the quasi-Painlevé property along a rectifiable curve, that is, for general solutions, every movable singularity defined by a rectifiable curve is at most an algebraic branch point. Moreover we discuss the global many-valuedness of their solutions. For the exceptional equation, by the method of successive approximation, we construct a general solution having a movable logarithmic branch point.

2000 Mathematics Subject Classification. Primary 34M55; Secondary 34M35.
Key words and phrases. Nonlinear differential equation, quasi-Painlevé property, Painlevé equation, hyperelliptic integral.

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