Blow up and near soliton dynamics for the $L^2$ critical gKdV equation

Martel, Yvan; Merle, Frank; Raphaël, Pierre

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 37, 14 p.

These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in $H^1$, construction of various exotic blow up rates in $H^1$, including grow up in infinite time.

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