Uniqueness of topological vortex in a self-dual Maxwell-Chern-Simons-Higgs system (Q2788637)

The paper addresses the system of coupled nonlinear elliptic equations for real fields \(u\) and \(N\) in the two-dimensional space: NEWLINE\[NEWLINE\begin{aligned} \Delta u &=2q(e^u - 1+\kappa N) +4\pi \delta(\mathbf{r} - \mathbf{r}_0), \\ \Delta N &=\kappa q^2(e^u - 1+\kappa N) +2qe^u N, \end{aligned}NEWLINE\]NEWLINE where \(\Delta\) is the two-dimensional Laplacian, \(\kappa\) and \(q\) are positive constants, \(\mathbf{r}_0\) is position of a singe vortex considered in the system, and \(\delta(\mathbf{r} - \mathbf{r}_0)\) is the Dirac's delta-function of the two-dimensional vectorial argument. The system is derived as a stationary nonrelativistic self-dual reduction of the known Maxwell-Chern-Simons-Higgs model of the classical field theory. The paper produces a rigorous proof of the uniqueness of the vortex solution of this system under certain conditions imposed on parameters (in particular, \(\kappa^2q > 2\)).