We consider the quintic generalized Korteweg–de Vries equation (gKdV) which is a canonical mass critical problem, for initial data in H1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see ,  and . For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in .
In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see , ,  and , for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L2 norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in .
"Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton." Acta Math. 212 (1) 59 - 140, 2014. https://doi.org/10.1007/s11511-014-0109-2