Pinning of magnetic vortices by an external potential

The existence and uniqueness of vortex solutions is proved for Ginzburg-Landau equations with external potentials in $\mathbb{R} ^2$. These equations describe the equilibrium states of superconductors and the stationary states of the $U(1)$-Higgs model of particle physics. In the former case, the external potentials are due to impurities and defects. Without the external potentials, the equations are translationally (as well as gauge) invariant, and they have gauge equivalent families of vortex (equivariant) solutions called magnetic or Abrikosov vortices, centered at arbitrary points of $\mathbb{R} ^2$. For smooth and sufficiently small external potentials, it is shown that for each critical point $z_0$ of the potential there exists a perturbed vortex solution centered near $z_0$, and that there are no other single vortex solutions. This result confirms the ``pinning'' phenomena observed and described in physics, whereby magnetic vortices are pinned down to impurities or defects in the superconductor.