# Vortex rings for the Gross-Pitaevskii equation

### Didier Smets

UPMC, Université Paris 06, France### Giandomenico Orlandi

Università di Verona, Italy### Fabrice Bethuel

Université Pierre et Marie Curie, Paris, France

## Abstract

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

## Cite this article

Didier Smets, Giandomenico Orlandi, Fabrice Bethuel, Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. 6 (2004), no. 1, pp. 17–94

DOI 10.4171/JEMS/2