Numerical Evidence for Stable Hopf Solitons in a Three-Dimensional Sigma-Skyrme Field Model

AbstractWe present a numerical investigation of toroidal soliton congurations in a three-dimensionalnonlinear vector eld model combining sigma and Skyrme-type contributions. The eld isdescribed by a unit vector n(x) ∈ S2 and is evolved on a cubic lattice using an energyrelaxation scheme.The simulations produce stable localized structures with toroidal energy density, con-sistent with the expected geometry of Hopf-type solitons. We examine the formation andstability of these congurations, their behavior under strong random perturbations, andtheir interaction in two-soliton setups.The results show robust relaxation towards stable toroidal congurations and weak re-pulsive interaction between separated structures. These ndings provide numerical evidencefor stable Hopf-type soliton congurations in this class of nonlinear eld models.