\{(x^1,x^2):  x^1 x^1+x^2 x^2=r^2 \}$ and $S_\infty^1=\lim_{r\to +\infty} S_r$, have a classically quantized magnetic flux: $\frac{1}{2\pi}\int_{\mathbb{R}^2}\, dx^1dx^2 \, F_{12}(x^1,x^2)=N\in\mathbb{Z}$.

It is clear from (\[asymptotic\]) that the vector field $A_i$ is asymptotically purely vorticial. Jaffe and Taubes, showed, see [@Jaffe1980], that the solutions are determined from $N$ points freely located in the $\mathbb{R}^2$ plane, around each of which the vector field $A_i$ is a quantized vortex, the total magnetic charge being equal to $N$. It is well known that these magnetically charged objects are also solutions of the second-order static field equations and, because they satisfy the PDE system (\[pde1\]), the BPS vortices are absolute minima of the action in the different topological sectors characterized by $N$. Therefore, these BPS or self-dual vortices are stable. The upper signs in (\[pde1\]) refer to the topological defects with a positive winding number, $N>0$, (vortices), of the map from the $\mathbb{R}^2$ boundary circle $S_\infty^1$ at infinity to the vacuum circle $S^1_1$ determined by the asymptotic behaviour of the complex field, see (\[asymptotic\]). Solutions of (\[pde1\]) with the lower signs are topological defects with negative winding number, $N<0$, (anti-vortices).

Without loss of generality, we shall focus in this paper on solutions with positive magnetic charge, 