Blow up and near soliton dynamics for the L 2 critical gKdV equation

These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation ut + (uxx + u)x = 0 for initial data in H close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in H, construction of various exotic blow up rates in H, including grow up in infinite time.