Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation

First, we show that there exist universal smooth profiles Qk∈S(R) (with Q0=Q) and a constant c0∈R such that, fixing the blow up time at t=0 and appropriate scaling and translation parameters, S satisfies, for any m⩾0,∂xmS(t)−∑k=0[m/2]1t12+m−2kQk(m−k)(⋅+1tt+c0)→0 in L2as t↓0. Second, we prove that, for 0