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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Existence and uniqueness of Tronquée solutions of the fourth-order Jimbo-Miwa second Painlevé equation


Authors: Nalini Joshi and Tegan Morrison
Journal: Proc. Amer. Math. Soc. 137 (2009), 2005-2014
MSC (2000): Primary 33E17, 34M55
Published electronically: January 16, 2009
MathSciNet review: 2480282
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Abstract: We consider the asymptotic limit as the independent variable approaches infinity, of the fourth-order second Painlevé equation obtained from a hierarchy based on the Jimbo-Miwa Lax pair. We prove that there exist two families of algebraic formal power series solutions and that there exist true solutions with these behaviours in sectors $ \sigma$ of the complex plane. Given $ \sigma$ we also prove that there exists a wider sector $ \Sigma \supset \sigma$ in which there exists a unique solution in each family. These provide the analogue of Boutroux's tri-tronquée solutions for the classical second Painlevé equation. Surprisingly, they also extend beyond the tri-tronquée solutions in the sense that we find penta-, hepta-, ennea-, and hendeca-tronquée solutions.


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Additional Information

Nalini Joshi
Affiliation: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia
Email: nalini@maths.usyd.edu.au

Tegan Morrison
Affiliation: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia
Email: teganm@maths.usyd.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09819-0
PII: S 0002-9939(09)09819-0
Received by editor(s): October 16, 2007
Received by editor(s) in revised form: April 9, 2008
Published electronically: January 16, 2009
Additional Notes: The authors gratefully acknowledge the support of the Australian Research Council through Discovery Grant DP0559019 and an Australian Postgraduate Award
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.