### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 065, 29 pages      arXiv:1704.04851      https://doi.org/10.3842/SIGMA.2017.065
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### Rational Solutions of the Painlevé-II Equation Revisited

Peter D. Miller and Yue Sheng
Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, USA

Received April 18, 2017, in final form August 07, 2017; Published online August 16, 2017

Abstract
The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlevé-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlevé-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method.

Key words: Painlevé equations; rational functions; Riemann-Hilbert problems; steepest descent method.

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