Numerical vortex solutions in (3+1) dimensions for the extended $C{P}^N$ Skyrme-Faddeev model

We construct numerical vortex solutions in a (3+1) dimensional Minkowski space-time for the extended version of the Skyrme-Faddeev model with target space $C{P}^N$. The solutions are essentially composed of $N$-th single vortex which does not belong to the integrable sector. They have finite energy per unit length and contain waves propagating along vortices with the speed of light. In this Letter we report on the case N=2 and show existence of the solutions with the charges $(n_1,n_2)=(1,2)$.