Existence and Uniqueness of BPS Vacuum and Multi-vortices in Inhomogeneous Abelian Higgs Model

TL;DR

This paper proves the existence and uniqueness of BPS vacuum and multi-vortex solutions in an inhomogeneous Abelian Higgs model, extending understanding of topological solitons in inhomogeneous media.

Contribution

It establishes rigorous mathematical results on the existence and uniqueness of BPS solutions in an inhomogeneous setting, which was previously unexplored.

Findings

Existence of zero-energy BPS vacuum solution.

Existence of quantized positive-energy multi-vortex solutions.

Uniqueness of these solutions under certain conditions.

Abstract

The BPS limit of the inhomogeneous abelian Higgs model is considered in $(1+2)$-dimensions. The second order Bogomolny equation is examined in the presence of an inhomogeneity expressed as a function of spatial coordinates. Assuming a physically reasonable upper bound on the $L^{2}(\mathbb{R}^{2})$ norm of the inhomogeneity function, we prove the existence and the uniqueness of nontrivial BPS vacuum solution of zero energy and topological BPS multi-vortex solutions of quantized positive energies.