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

Yuanyang Hu

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.

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

Abstract

Let be a connected finite graph. We investigate two Bogomol'nyi-Prased-Sommerfeld equations on . We establish necessary and sufficient conditions for the existence and uniqueness of solutions to the two BPS equations.
Paper Structure (4 sections, 8 theorems, 111 equations)

This paper contains 4 sections, 8 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Settings and Main Results
  3. The proof of Theorem \ref{['t1']}
  4. The proof of Theorem \ref{['t2']}

Key Result

Theorem 2.1

Equations 11 admits a unique solution if and only if $\max\limits_{1\le j \le l} \{N_j \} < \frac{(l+1)}{4 \pi} |V|$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['t1']}
  • Lemma 4.1
  • ...and 4 more