Constructive Approximation

, Volume 44, Issue 2, pp 195–231 | Cite as

Asymptotic Behavior of the Fourth Painlevé Transcendents in the Space of Initial Values

Article

Abstract

We study the asymptotic behavior of solutions of the fourth Painlevé equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.

Keywords

The fourth Painlevé equation Asymptotic behavior Resolution of singularities Rational surface Space of initial values 

Mathematics Subject Classification

34M55 34M30 

Notes

Acknowledgments

The research reported in this paper was supported by Grant No. FL120100094 from the Australian Research Council. The work of M.R. was partially supported by Project No. 174020: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems of the Serbian Ministry of Education, Science and Technological Development. The authors are grateful to the referee for comments and questions, which led to improvement of the manuscript. M.R. thanks Viktoria Heu for useful discussions.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

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

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