Title: Rational Solutions of the Painlevé-II Equation Revisited,
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.