Rational solutions of Painlevé-II equation as Gram determinant

Under the Flaschka–Newell Lax pair, the Darboux transformation for the Painlevé-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.