  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.
