Non-linear superposition law and Skyrme crystals

Exact configurations of the four dimensional Skyrme model are presented. The static configurations have the profile which behaves as a kink and, consequently, the corresponding energy momentum tensor describes a domain wall. Furthermore, a class of exact time periodic Skyrmions is discovered. Within such class, it is possible to disclose a remarkable phenomenon which is a genuine effect of the Skyrme term. For a special value of the frequency the Skyrmions admit a non linear superposition principle. One can combine two or more exact elementary Skyrmions (which may depend in a non trivial way on all the space like coordinates) into a new exact composite Skyrmion. Due to such superposition law, despite the explicit presence of non linear effects in the energy momentum tensor, the interaction energy between the elementary Skyrmions can be computed exactly. The relations with the appearance of Skyrme crystals is discussed.