Near soliton dynamics and singularity formation for $L^2$ critical   problems
  This survey reviews the state of the art concerning the singularity formation for two canonical dispersive problems: the mass critical non linear Schr\"odinger equation and the mass critical generalized KdV equation. In particular, we address the question of the classification of the flow for initial data near the soliton. 
Yvan Martel, Frank Merle, Pierre Raphael, Jeremie Szeftel
math.AP