Near-BPS baby Skyrmions with Gaussian tails

We consider the baby Skyrme model in a physically motivated limit of reaching the restricted or BPS baby Skyrme model, which is a model that enjoys area-preserving diffeomorphism invariance. The perturbation consists of the kinetic Dirichlet term with a small coefficient $\epsilon$ as well as the standard pion mass term, with coefficient $\epsilon m_1^2$. The pions remain lighter than the soliton for any $\epsilon$ and therefore the model is physically acceptable, even in the $\epsilon \to 0$ limit. The version of the BPS baby Skyrme model we use has BPS solutions with Gaussian tails. We perform full numerical computations in the $\epsilon\to 0$ limit and even reach the strict $\epsilon=0$ case, finding new nontrivial BPS solutions, for which we do not yet know the analytic form.