Dispersionless integrable systems and the Bogomolny equations on an Einstein-Weyl geometry background

TL;DR

This paper develops a dispersionless integrable system linked to three-dimensional Einstein-Weyl geometries, constructs its matrix extension, and connects it to the Bogomolny equations for non-abelian monopoles, expanding the understanding of integrable structures in geometric backgrounds.

Contribution

It introduces a novel dispersionless integrable system for Einstein-Weyl geometries, along with its matrix extension and a dressing scheme, relating to Bogomolny equations.

Findings

Derived a dispersionless integrable system for Einstein-Weyl geometry

Constructed the matrix extension of the integrable system

Connected the system to Bogomolny equations for monopoles

Abstract

We derive a dispersionless integrable system describing a local form of a general three-dimensional Einstein-Weyl geometry with an Euclidean (positive) signature, construct its matrix extension and demonstrate that it leads to the Bogomolny equations for a non-abelian monopole on an Einstein-Weyl geometry background. The corresponding dispersionless integrable hierarchy, its matrix extension and the dressing scheme are also considered.