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Tohoku Mathematical Journal
2008
December
SECOND SERIES VOL. 60, NO. 4
Tohoku Math. J.
60 (2008), 581595

Title
NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND PAINLEVÉ TYPE WITH THE QUASIPAINLEVÉ 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 quasiPainlevé 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 manyvaluedness 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, quasiPainlevé property, Painlevé equation, hyperelliptic integral.


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