Some vortex solutions in the extended Skyrme-Faddeev model

Analytical and numerical vortex solutions for the extended Skyrme-Faddeev model in a (3 + 1) dimensional Minkowski space-time are investigated. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction of the solutions has been done in twofold: one makes use of an axially symmetric ansatz and solves the resulting ODE by an analytical and a numerical way. The analytical vortices are obtained for special form of the potentials, and the numerical ones are computed using the successive over relaxation method for wider choice of the potentials. Another is based on a simulational technique named the simulated annealing method which is available to treat the non-axisymmetric shape of solutions. The crucial thing for determining the structure of vortices is the type of the potential.