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Systems with Coulomb interactions

Sylvia Serfaty (2014)

Journées Équations aux dérivées partielles

Systems with Coulomb and logarithmic interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. In this review, we describe...

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 ).

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: d d t u + i - Δ + V ( x ) u = ν Δ u - γ R | u | 2 p u - i γ I | u | 2 q u + ν η ( t , x ) . ( * ) The force η is white in time and smooth in x ; the potential V ( x ) is typical. We are concerned with the limiting, as ν 0 , behaviour of solutions on long time-intervals 0 t ν - 1 T , and with behaviour of these solutions under the double limit t and ν 0 . We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for(...

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