Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

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 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 [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

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 [31], [39], [32] and [33], 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 $$\Vert u_x{(t)\Vert }_{L^2}\sim \frac{C(u_0)}{T-t}\mathrm{as}t\to T.$$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 [31].