= Logarithmic equation for inverse-density Abelian Higgs vortices
{title2=$\Delta u+e^u(1-e^u)=4\pi\sum_jn_j\delta_{p_j}$}
Write $\Phi=e^{u/2+i\chi}$ away from its zeros and use the oriented planar <Hodge star> with $*dx^1=dx^2$. The first <Bogomolny equation> gives $A=d\chi-\tfrac12*du$. A zero of multiplicity $n_j$ contributes $u=2n_j\log|x-p_j|+\text{smooth}$ and phase winding $2\pi n_j$. Combining the smooth magnetic field with these delta sources gives the displayed equation and the topological boundary condition $u\to0$. Prescribed zero positions and multiplicities reduce multi-vortex construction to this scalar elliptic problem; a regular solution reconstructs both smooth fields.
Back to article page
