On the long time behavior of KdV type equations

Tzvetkov, Nikolay

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 933, pp. 219-248.

Dans une série d'articles récents, Martel et Merle ont mis en évidence l'existence de solutions qui explosent en temps fini, dans l'espace d'énergie, pour l'équation de KdV généralisée critique, résolvant ainsi une conjecture ancienne. Ils ont introduit des outils nouveaux pour étudier la dynamique non linéaire au voisinage d'une onde solitaire. Le but de cet exposé est de présenter les idées principales développées par Martel-Merle.

In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg-de-Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.

Classification : 35Q53, 35B35
Keywords: explosion en temps fini, EDP hamiltonienne, KdV
Mot clés : blow-up solutions, hamiltonian PDE, KdV

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