Symmetric vortices for two-component Ginzburg-Landau systems

TL;DR

This paper investigates symmetric vortex solutions in two-component Ginzburg-Landau systems, establishing their existence, uniqueness, asymptotic behavior, and monotonicity properties through analytical methods.

Contribution

It provides the first comprehensive analysis of symmetric vortex solutions, including existence, uniqueness, and detailed monotonicity regimes for two-component Ginzburg-Landau equations.

Findings

Existence and uniqueness of symmetric vortex solutions for large radii.

Identification of parameter regimes with monotone and non-monotone vortex profiles.

Asymptotic behavior characterization of vortex solutions at infinity.

Abstract

We study Ginzburg--Landau equations for a complex vector order parameter Psi=(psi_+,psi_-). We consider symmetric (equivariant) vortex solutions in the plane R^2 with given degrees n_\pm, and prove existence, uniqueness, and asymptotic behavior of solutions for large r. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles |psi_+|, |psi_i| are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and supersolution construction and a comparison theorem for elliptic systems.