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

TL;DR

This paper establishes the precise conditions under which solutions to the Bogomol'nyi equations exist and are unique on finite connected graphs, extending the understanding of these equations from continuous to discrete graph settings.

Contribution

It provides a necessary and sufficient condition for the existence and uniqueness of solutions to the Bogomol'nyi equations on finite graphs, a novel extension from continuous domains.

Findings

Derived a condition for solution existence on graphs

Proved the uniqueness of solutions under certain conditions

Extended Bogomol'nyi equations analysis to discrete graph structures

Abstract

Let $G=(V,E)$ be a connected finite graph. We study the Bogomol'nyi equation \begin{equation*} \Delta u= \mathrm{e}^{u}-1 +4 \pi \sum_{s=1}^{k} n_s \delta_{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.