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. 
Kyrylo Andreiev, Iryna Egorova, Till Luc Lange, and Gerald Teschl
nlin.SI math-ph math.AP math.MP