Existence and uniqueness of solutions to Bogomol'nyi equations on graphs

Abstract

Let $G=(V,E)$ be a connected finite graph. We study the Bogomol'nyi equation \begin{equation*} Δu= \mathrm{e}^{u}-1 +4 π\sum_{s=1}^{k} n_s δ_{z_{s}} \quad \text { on } \quad G, \end{equation*} where $z_1,z_2,\dots ,z_k$ are arbitrarily chosen distinct vertices on the graph, $n_j$ is a positive integer, $j=1,2,\cdots ,k$ and ${\delta }_{z_s}$ is the Dirac mass at $z_s$. We obtain a necessary and sufficient condition for the existence and uniqueness of solutions to the Bogomol'nyi equation.

arXiv comment13 pages