Long time behavior of solutions to the mKdV
  In this paper we consider the long time behavior of solutions to the modified Korteweg-de Vries equation on R. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotics without relying on complete integrability. We also consider the asymptotic completeness problem. Our result uses the method of testing by wave packets, developed in the work of Ifrim and Tataru on the 1d cubic nonlinear Schr\"odinger and 2d water wave equations. 
Benjamin Harrop-Griffiths
math.AP