Authors’ abstract: We consider the quintic generalized Korteweg-de Vries equation (gKdV) \[ u_t +(u_{xx}+u^5)_x=0, \] which is a canonical mass critical problem, for initial data in \(H^{1}\) 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. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [Y. Martel and F. Merle, J. Am. Math. Soc. 15, No. 3, 617–664 (2002; Zbl 0996.35064)].
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. 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 \(L^2\) norm; (ii) the solution is global and converges to a soliton as \(t\to\infty\); (iii) the solution blows up in finite time \(T\) with speed \[ \|u_x(t)\| _{L^2}\sim\frac{C(u_0)}{T-t}\quad\text{as }t\to T. \]
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 [F. Merle and P. Raphaël, J. Am. Math. Soc. 19, No. 1, 37–90 (2006; Zbl 1075.35077)].