Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg–de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.