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Habilitation à diriger des recherches

Quelques contributions à l'analyse mathématique de l'équation de Gross-Pitaevskii et du modèle de Bogoliubov-Dirac-Fock

Abstract : This dissertation is devoted to the mathematical analysis of the Gross-Pitaevskii equation on one hand, and of the Bogoliubov-Dirac-Fock model on the other hand. Concerning the Gross-Pitaevskii equation, the analysis begins by the variational construction of the minimizing travelling waves. This construction is completed by the proof of the orbital stability of the black soliton in dimension one, as well as the description of the transonic limit of the minimizing travelling waves towards the solitary waves for the Kadomtsev-Petviashvili equation in dimension two. The rigorous derivation of the long wave regime towards the Korteweg-de Vries equation in dimension one concludes the analysis. The description of the Bogoliubov-Dirac-Fock model focuses on the construction of the ground states for the reduced model. Their charge renormalization next provides an asymptotic expansion of the charge density of the polarized vacuum, which agrees with the perturbative expansions given by the quantum electrodynamics.
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Habilitation à diriger des recherches
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https://tel.archives-ouvertes.fr/tel-00706916
Contributor : Philippe Gravejat Connect in order to contact the contributor
Submitted on : Monday, June 11, 2012 - 5:03:58 PM
Last modification on : Thursday, March 5, 2020 - 6:19:46 PM
Long-term archiving on: : Wednesday, September 12, 2012 - 2:41:14 AM

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Philippe Gravejat. Quelques contributions à l'analyse mathématique de l'équation de Gross-Pitaevskii et du modèle de Bogoliubov-Dirac-Fock. Equations aux dérivées partielles [math.AP]. Université Paris Dauphine - Paris IX, 2011. ⟨tel-00706916⟩

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