Vortex Ring Pinning for the Gross-Pitaevskii Equation in Three-Dimensional Space

We construct stationary solutions possessing two vortex rings to the nonlinear Schrödinger type problem iut = −ε4u + ( V + |u| ) u, where the unknown function u is defined as u : R3×R→ C, ε is a small positive parameter and V is a real smooth potential. These two vortex rings will be pinned at a fixed site by the potential V . They lie in the same plane and have neighboring interaction in the normal direction, or in two parallel planes with interaction in the binormal direction, in such a way that the neighboring interaction will be balanced by the effect of the potential.