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Improved estimates for the Ginzburg-Landau equation : the elliptic case

Fabrice Bethuel;  Giandomenico Orlandi;  Didier Smets — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the G L -energy E ε and the parameter ε . These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel;  G. Orlandi;  Didier Smets — 2004

Journal of the European Mathematical Society

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N ≥ 3 . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3 ).

Ondes progressives pour l’équation de Gross-Pitaevskii

Fabrice Béthuel;  Philippe Gravejat;  Jean-Claude Saut

Séminaire Équations aux dérivées partielles

Cet exposé présente les résultats de l’article [] au sujet des ondes progressives pour l’équation de Gross-Pitaevskii : la construction d’une branche d’ondes progressives non constantes d’énergie finie en dimensions deux et trois par un argument variationnel de minimisation sous contraintes, ainsi que la non-existence d’ondes progressives non constantes d’énergie petite en dimension trois.

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