tensor $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$. We choose a systems of units where $c=1$, but $\hbar$ has dimensions of length$\times$mass. The metric tensor in the Minkowski space $\mathbb{R}^{(2,1)}$ is chosen as $g_{\mu\nu}={\rm diag}(1,-1,-1)$ with $\mu,\nu=0,1,2$. The parameter $\kappa^2=\frac{\lambda}{e^2}$, where $\lambda$ is the quartic self-coupling of the scalar field, measures the ratio between the square of the masses of the Higgs, $M^2=\lambda v^2$, and the vector particles, $m^2=e^2 v^2$. Bogomolnyi-Prasad-Sommerfield (self-dual) vortices arise when the parameter $\kappa^2$ is set to unity, $\kappa^2=1, $ in the action (\[action1\]). These vortices are solitonic topological defects (static and spatially localized solutions of the field equations) for which the static energy density functional $$V[\phi,A]=v^2\int d^2 x \Big[ \frac{1}{4} F_{ij} F_{ij} + \frac{1}{2} (D_i\phi)^* D_i\phi + \frac{1}{8} (\phi^*\phi-1)^2 \Big]$$ is finite. A Bogomolnyi arrangement of $V[\phi,A]$ $$V[\phi,A]=\frac{v^2}{2}\int_{\mathbb{R}^2} \, d^2x \, \left\{\Big(F_{12}\pm\frac{1}{2}(\phi^*\phi-1)\Big)^2+\big\vert D_1\phi\pm i D_2 \phi\vert^2\right\}+\frac{v^2}{2} \,\,\Big\vert \int_{\mathbb{R}^2} \, d^2x \, F_{12}\Big\vert \label{bogs}$$ leads us to conclude that solutions of the first-order PDE system $$D_1\phi \pm i D_2 \phi =0 \hspace{0.5cm},\hspace{0.5cm} F_{12}\pm \frac{1}{2} (\phi^*\phi -1)=0 \label{pde1}$$ complying with the asymptotic boundary conditions $$\phi^* \phi|_{S_\infty^1}=1 \hspace{0.5cm} \mbox{and} \hspace{0.5cm} D_i\phi|_{S_\infty^1}=0 \hspace{0.5cm} \equiv \hspace{0.5cm}
 \phi |_{S^1_\infty}=e^{i N \theta} \hspace{0.5cm} \mbox{and} \hspace{0.5cm} A_i|_{S^1_\infty}=-i N\phi^*\partial_i \phi|_{S^1_\infty} \label{asymptotic}$$ where $\theta =\arctan \frac{x^2}{x^1}$, $S_r^1= 