Existence and uniqueness of Tronquée solutions of the fourth-order Jimbo-Miwa second Painlevé equation
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- by Nalini Joshi and Tegan Morrison PDF
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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.References
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Additional Information
- Nalini Joshi
- Affiliation: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia
- MR Author ID: 248776
- ORCID: 0000-0001-7504-4444
- 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2005-2014
- MSC (2000): Primary 33E17, 34M55
- DOI: https://doi.org/10.1090/S0002-9939-09-09819-0
- MathSciNet review: 2480282