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

TL;DR

This paper studies the existence and uniqueness of solutions to two Bogomol'nyi-Prased-Sommerfeld equations on finite graphs, providing necessary and sufficient conditions for their solutions.

Contribution

It establishes the first comprehensive criteria for the existence and uniqueness of solutions to these equations on finite graphs.

Findings

Necessary and sufficient conditions for solutions

Uniqueness of solutions on finite graphs

Framework applicable to related equations

Abstract

Let $G=(V,E)$ be a connected finite graph. We investigate two Bogomol'nyi-Prased-Sommerfeld equations on $G$. We establish necessary and sufficient conditions for the existence and uniqueness of solutions to the two BPS equations.