# Rational Solutions of the Painlev\'e-II Equation Revisited

## TL;DR

This paper reviews the algebraic properties of rational solutions to the Painlevé-II equation, explores three Riemann-Hilbert representations, and discusses their connections and asymptotic analysis using the steepest descent method.

## Contribution

It compares and connects three different Riemann-Hilbert representations of rational Painlevé-II solutions and reviews recent asymptotic results derived from these frameworks.

## Key findings

- Explicit connection between Flaschka-Newell and Bertola-Bothner representations.
- Descriptions of asymptotic behavior via steepest descent method.
- Clarification of algebraic and analytic properties of rational solutions.

## Abstract

The rational solutions of the Painlev\'e-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 Painlev\'e-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 Painlev\'e-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlev\'e-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method.