Symmetric vortices for two-component Ginzburg–Landau systems

We study Ginzburg–Landau equations for a complex vector order parameter Ψ=(ψ+,ψ−)∈C2. We consider symmetric vortex solutions in the plane R2, ψ(x)=f±(r)ein±θ, with given degrees n±∈Z, and prove the existence, uniqueness, and asymptotic behavior of solutions as r→∞. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles f+, f− are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and super-solution construction and a comparison theorem for elliptic systems.