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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic behaviour of the third Painlevé transcendents in the space of initial values


Authors: Nalini Joshi and Milena Radnović
Journal: Trans. Amer. Math. Soc. 372 (2019), 6507-6546
MSC (2010): Primary 34M55; Secondary 34M30
DOI: https://doi.org/10.1090/tran/7899
Published electronically: July 31, 2019
MathSciNet review: 4024529
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Abstract: We study the asymptotic behaviour of the solutions of the generic ($ D_6^{(1)}$-type) third Painlevé equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with an essential singularity at infinity has an infinite number of poles and zeros, and a we obtain a similar result at the origin.


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Additional Information

Nalini Joshi
Affiliation: School of Mathematics and Statistics F07, University of Sydney, New South Wales 2006, Australia
Email: nalini.joshi@sydney.edu.au

Milena Radnović
Affiliation: School of Mathematics and Statistics F07, University of Sydney, New South Wales 2006, Australia
Email: milena.radnovic@sydney.edu.au

DOI: https://doi.org/10.1090/tran/7899
Received by editor(s): January 24, 2018
Received by editor(s) in revised form: January 16, 2019
Published electronically: July 31, 2019
Additional Notes: This research was supported by Australian Laureate Fellowship #FL120100094 from the Australian Research Council.
The research of the second author was partially supported by the Serbian Ministry of Education and Science (Project #174020: Geometry and Topology of Manifolds and Integrable Dynamical Systems).
Article copyright: © Copyright 2019 American Mathematical Society