Rational solutions of Painlev\'{e}-II equation as Gram determinant

TL;DR

This paper constructs rational solutions to the Painlevé-II equation using Darboux transformations, represents them via Gram determinants, analyzes their asymptotics, and connects these solutions to a Riemann-Hilbert problem.

Contribution

It introduces a novel Darboux transformation approach for Painlevé-II, expressing rational solutions as Gram determinants and deriving their asymptotic behavior.

Findings

Rational solutions expressed as Gram determinants

Large y asymptotics of solutions obtained

Connection to Riemann-Hilbert problem established

Abstract

Under the Flaschka-Newell Lax pair, the Darboux transformation for the Painlev\'{e}-II equation is constructed by the limiting technique. With the aid of the Darboux transformation, the rational solutions are represented by the Gram determinant, and then we give the large $y$ asymptotics of the determinant and the rational solutions. Finally, the solution of the corresponding Riemann-Hilbert problem is obtained from the Darboux matrices.