Constructive Approximation

, Volume 39, Issue 1, pp 11–41 | Cite as

Global Asymptotics of the Second Painlevé Equation in Okamoto’s Space



We study the solutions of the second Painlevé equation (P II) in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlevé equation (P 34). By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repeller for the dynamics in Okamoto’s space. Moreover, we show that the limit set of the solutions exists and is compact and connected.


The second Painlevé equation Thirty-fourth Painlevé equation Asymptotic behavior Resolution of singularities Rational surface 

Mathematics Subject Classification (2000)

34M55 34E05 34M55 34M30 14E15 



This research was supported by an Australian Postgraduate Award and by the Australian Research Council grant 3DP110102001.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and Statistics F07The University of SydneySydneyAustralia

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