Five-dimensional Monopole Equation with Hedge-Hog Ansatz and Abel's Differential Equation

TL;DR

This paper reviews five-dimensional monopoles, explores a numerical Hedge-Hog solution, and transforms the Bogomol'nyi equation into Abel's differential equation, providing insights into higher-dimensional gauge theories.

Contribution

It introduces a novel approach to solving five-dimensional monopole equations by translating them into Abel's differential equation, offering new analytical and numerical methods.

Findings

Numerical Hedge-Hog solutions are obtained.

The Bogomol'nyi equation is reformulated as Abel's differential equation.

The approach advances understanding of higher-dimensional monopoles.

Abstract

We review the generalized monopole in the five-dimensional Euclidean space. A numerical solution with the Hedge-Hog ansatz is studied. The Bogomol'nyi equation becomes a second order autonomous non-linear differential equation. The equation can be translated into the Abel's differential equation of the second kind and is an algebraic differential equation.