The existence of Bogomolny decomposition for baby Skyrme models

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.

💡 Research Summary

The paper investigates the existence of Bogomolny decompositions for baby Skyrme models in two spatial dimensions (2+0). Two variants are considered: the full baby Skyrme model, which contains both the quadratic sigma‑model term and the quartic Skyrme term together with an arbitrary potential V(φ), and the restricted (or pure) baby Skyrme model, which retains only the quartic term and the potential. The authors employ the method of “strong necessary conditions” – a variational technique that replaces the usual Euler‑Lagrange equations with a set of sufficient first‑order conditions obtained by acting with differential operators on the Lagrangian density. This approach is designed to produce first‑order Bogomolny equations even when the potential is highly non‑linear.

For the restricted model the Lagrangian density is ℒ_R = λ(∂_i φ·∂_j φ)^2 – V(φ). Applying the strong necessary‑condition formalism yields a universal first‑order relation
∂1 φ = ± ε{ij} ∂_j φ f(φ), with f(φ)=√