Exact results in the Skyrme model in (3 + 1) dimensions via the generalized hedgehog ansatz

We present exact results in the (3 + 1)-dimensional Skyrme model. First of all, it will be shown that, in the Pionic sector, a quite remarkable phenomenon for a non-integrable (3+1)-dimensional field theory appears: a non-linear superposition law is available allowing the composition of solutions in order to generate new solutions of the full field equations keeping alive, at the same time, the interactions terms in the energy-density. Secondly, it will be shown that the generalized hedgehog ansatz can be extended to suitable curved backgrounds. Interestingly, one can choose the background metric in such a way to describe finite-volume effects and, at the same time, to simplify the Skyrme field equations. In this way, it is possible to construct the first exact multi-Skyrmionic configurations of the (3 + 1)-dimensional Skyrme model with arbitrary high winding number and living at finite volume. Last but not least, a novel BPS bound (which is sharper than the usual one in term of the winding number) will be derived which can be saturated and reduces the field equations to a first-order equation for the profile