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

Let S be a minimal mass blow up solution of the critical generalized KdV equation as constructed in [25]. We prove both time and space sharp asymptotics for S close to the blow up time. Let Q be the unique ground state of (gKdV), satisfying Q" + Q(5) = Q. First, we show that there exist universal smooth profiles Q(k) is an element of S(R) (with Q(0) = Q) and a constant c(0) is an element of R such that, fixing the blow up time at t = 0 and appropriate scaling and translation parameters, S satisfies, for any m >= 0, partial derivative(m)(x) s(t)- Sigma([m/2])(k=0) 1/t(1/2)+m-2k Q(K)((m-k)) (.+ 1/t/t +c(0) )-> 0 in L-2 as t down arrow 0. second,we prove that, for 0 < t << 1, x <= -1/t-1, s(t,x)similar to-1/2\\Q\\ L-1 \x\(-3/2,) and related bounds for the derivatives of S(t) of any order. We also prove integral S-R(t,x) dx = 0. (C) 2017 Elsevier Masson SAS. All rights reserved.