Near-BPS baby Skyrmions

We consider the baby-Skyrme model in the regime close to the so-called restricted baby-Skyrme model, which is a BPS model with area-preserving diffeomorphism invariance. The perturbation takes the form of the standard kinetic Dirichlet term with a small coefficient $\epsilon$. Classical solutions of this model, to leading order in $\epsilon$, are called restricted harmonic maps. In the BPS limit ($\epsilon\to 0$) of the model with the potential being the standard pion-mass term, the solution with unit topological charge is a compacton. Using analytical and numerical arguments we obtain solutions to the problem for topological sectors greater than one. We develop a perturbative scheme in $\epsilon$ with which we can calculate the corrections to the BPS mass. The leading order ($\mathcal{O}(\epsilon^1)$) corrections show that the baby Skyrmion with topological charge two is energetically preferred. The binding energy requires us to go to the third order in $\epsilon$ to capture the relevant terms in perturbation theory, however, the binding energy contributes to the total energy at order $\epsilon^2$. We find that the baby Skyrmions - in the near-BPS regime - are compactons of topological charge two, that touch each other on their periphery at a single point and with orientations in the attractive channel.