Radial solutions for the Ginzburg–Landau equation with applied magnetic field Original Research Article

The purpose of this work is a systematic study of symmetric vortices for the Ginzburg–Landau model of superconductivity along a cylinder, with applied magnetic field parallel to its axis. The Ginzburg–Landau constant κ of the material and the degree d of the vortex are fixed. For any given parameters r̄ (the radius of the cylinder) and h (the intensity of the applied magnetic field), one can find a symmetric vortex (ψ,A) which satisfies the boundary conditions View the MathML source and View the MathML source. It is then shown that symmetric vortices form a family depending continuously on two real parameters α and c which describe the behaviour at the center of the vortex. As the boundary conditions depend smoothly on those parameters, one can distinguish two main connected domains of vortices: the first one, defined by the boundary conditions View the MathML source and View the MathML source, is a zone of stability where |ψ| remains increasing; the second one, defined by the boundary conditions View the MathML source and View the MathML source, is a zone where some instability can occur. Attention is focused on the smooth branch of vortices which separates those two domains: it is indexed by the parameter α running from some L>0 to +∞, and the limit as α decreases to L corresponds to a Berger and Chen type vortex.