The existence of Bogomolny decomposition for baby Skyrme models

TL;DR

This paper derives Bogomolny equations for both full and restricted baby Skyrme models in two dimensions, revealing conditions under which these decompositions exist, with implications for understanding soliton solutions.

Contribution

It introduces a method to derive Bogomolny decompositions for baby Skyrme models, showing their existence depends on the potential form and model restrictions.

Findings

Bogomolny decomposition exists for the restricted baby Skyrme model with any potential.

For the full baby Skyrme model, such decomposition is only possible for specific potentials.

The method used is based on strong necessary conditions, providing a new approach to analyze these models.

Abstract

We derive the Bogomolny decompositions (Bogomolny equations) for: full baby Skyrme model and for its restricted version (so called, pure baby Skyrme model), in (2+0) dimensions, by using so called, concept of strong necessary conditions. It turns out that Bogomolny decomposition can be derived for restricted baby Skyrme model for arbitrary form of the potential term, while for full baby Skyrme model, such derivation is possible only for some class of the potentials.