Uniqueness of Symmetric Vortex Solutions in the Ginzburg–Landau Model of Superconductivity

Symmetric vortices are finite energy solutions ψ, A to the Ginzburg–Landau equations of superconductivity with the form ψ=f(r) eidθ, A=S(r)/r2(−y, x). The existence, regularity, and asymptotic form of the solutions f(r), S(r) for any d∈Z\{0} have been established by Plohr and by Burger and Chen. In this paper we prove the uniqueness of these solutions when the Ginzburg–Landau parameter κ satisfies κ2⩾2d2, for any fixed d∈Z\{0}. To do this, we show that any such solution is a non-degenerate relative minimizer of the free energy functional constrained to a convex set, then use a version of the Mountain Pass Theorem to derive a contradiction, should there be more than one solution.