Uniqueness of symmetric vortex solutions in the Ginzburg-Landau model of superconductivity

Symmetric vortices are finite energy solutions psi, A to the Ginzburg-Landa u equations of superconductivity with the form psi = f(r) e(id0), A = S(r)/ r(2)( -- y, x). The existence, regularity, and asymptotic form of the solut ions f(r), S(r) For any d epsilon Z\{0} have been established by Plohr and by Burger and Chen. In this pager we prove the uniqueness of these solution s when the Ginzburg-Landau parameter satisfies k(2) greater than or equal t o 2d(2), for any fixed d epsilon Z\{0}. To do this, we show that ally such solution is a non-degenerate relative minimizer of the free energy function al constrained to a convex set, then use a version of the Mountain Pass The orem to derive a contradiction, should there be more than one solution. (C) 1999 Academic Press.