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...

For external magnetic field hex ≤ Cε–α, we prove that a Meissner state solution for the Chern-Simons-Higgs functional exists. Furthermore, if the solution is stable among all vortexless solutions, then it is unique.

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N\ge 3$. 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 $N=3$).

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: $$\frac{\mathrm{d}}{\mathrm{d}t}u+\mathrm{i}\left(-\Delta +V\left(x\right)\right)u=\nu \left(\Delta u-{\gamma }_R{\left|u\right|}^{2p}u-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u\right)+\sqrt{\nu }\eta (t,x).(*)$$ The force $\eta$ is white in time and smooth in $x$; the potential $V\left(x\right)$ is typical. We are concerned with the limiting, as $\nu \to 0$, behaviour of solutions on long time-intervals $0\le t\le {\nu }^{-1}T$, and with behaviour of these solutions under the double limit $t\to \infty$ and $\nu \to 0$. We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for(...