Long-Time Asymptotics for 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 for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method. 
Katrin Grunert and Gerald Teschl