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

By Yvan Martel, F. Merle, Pierre Raphael. 2014

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AbstractWe consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ut+(uxx+u5)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 $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$‖ux(t)‖L2∼C(u0)T-tast→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].

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/af65cfe2af9482f4c0ee1cb38623056e1e9db2b5

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Citations in the corpus

Blow up for the critical gKdV equation II: minimal mass dynamics (2012, methodology): “In [29], we have proved the following (see [29, Theorems 1.”