On Bogomolny Decompositions for the Baby Skyrme 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.

The Existence of Bogomolny Decompositions for Gauged $O(3)$ Nonlinear ``sigma'' Model and for Gauged Baby Skyrme Models

“…We obtained analogical results for the gauged full baby Skyrme model in (2+0)-dimensions, in this case the Bogomolny decomposition has the form (58) and the potential needs to satisfy the condition (57). We see that analogically to [20], where Bogomolny decomposition for ungauged baby Skyrme models: restricted and full one, have been derived, the set of the solutions of Bogomolny decomposition of gauged full baby Skyrme model is the subset of the solutions of Bogomolny decomposition of gauged restricted baby Skyrme model. Moreover, at the beginning of this paper, we have assumed that for the gauged full baby Skyrme model and for the gauged restricted baby Skyrme model, the potentials in their hamiltonians, depend on ω, ω * , A 1 , A 2 and u, v, A 1 , A 2 , respetively.…”

“…Now, we consider ω, ω * , A i , (i = 1, 2), G k , (k = 1, 2, 3), as equivalent dependent variables, governed by the system of equations (20) - (31). We make two operations (similar operations were made firstly in [20], in the cases of ungauged baby Skyrme models: full and restricted one).…”

“…There was also showed that in the case of ungauged restricted baby Skyrme model, Bogomolny decomposition existed for arbitrary potential (in [9] Bogomolny equations had been obtained for the potential, which was a square of some non-negative function with isolated zeroes, but by another way than used in [20]). Next, in [20], it was also showed that for the case of ungauged full baby Skyrme model, the set of the solutions of corresponding Bogomolny equations was some subset of the set of the solutions of Bogomolny equations for ungauged restricted baby Skyrme model. The technique used in [19], for derivation of Bogomolny equations for gauged restricted baby Skyrme model in (2+0)-dimensions (in the case V (S i ) = 1 − n · S, (i = 1, 2, 3)), was firstly applied by Bogomolny in [21], among others, for the nonabelian gauge theory.…”

“…In [20], Bogomolny decompositions for both ungauged models: restricted and full baby Skyrme one, were derived. There was also showed that in the case of ungauged restricted baby Skyrme model, Bogomolny decomposition existed for arbitrary potential (in [9] Bogomolny equations had been obtained for the potential, which was a square of some non-negative function with isolated zeroes, but by another way than used in [20]).…”

Bogomolny equations for the BPS Skyrme models with impurity

“…Here let us mention the Abelian Higgs model at the critical coupling, 't Hooft-Polyakov monopole or Yang-Mills instantons as the most prominent examples. BPS property indicates for a given model that there is possible to derive Bogomolny equations for this model -such equations were derived for restricted BPS baby Skyrme model, [7,8], for gauged restricted BPS baby Skyrme model, [9]. Some aspects of relations between supersymmetry and Bogomolny equations are considered in [10][11][12].…”