Vortex Solutions of the High-κ High-Field Ginzburg-Landau Model with an Applied Current

In this paper, we study the single-vortex solutions of a two-dimensional high-κ high-field Ginzburg-Landau model of superconductivity with a constant applied current. Under a nondegeneracy condition and for appropriate ranges of the applied magnetic field and applied current, we construct some special solutions which, up to a constant shift of phase in time, are the stationary solutions of the model equation. Our result provides partial justification to the existence of a critical applied current which is the one important step towards a rigorous mathematical characterization of the interactions between the quantized vortices and applied electric current.