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2014 Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton
Yvan Martel, Frank Merle, Pierre Raphaël
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Acta Math. 212(1): 59-140 (2014). DOI: 10.1007/s11511-014-0109-2

Abstract

We consider the quintic generalized Korteweg–de Vries equation (gKdV) 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 ux(t)L2C(u0)T-tastT.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].

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Yvan Martel. Frank Merle. Pierre Raphaël. "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

Information

Received: 13 February 2012; Published: 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1301.35137
MathSciNet: MR3179608
Digital Object Identifier: 10.1007/s11511-014-0109-2

Rights: 2014 © Institut Mittag-Leffler

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Vol.212 • No. 1 • 2014
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