Vortex solutions of the nonlinear optical Maxwell–Dirac equations

We apply the method of slowly varying amplitudes of the electric and magnetic fields to the integro-differential system of nonlinear Maxwell equations. The equations are reduced to a system of differential nonlinear Maxwell amplitude equations (NME). Different orders of dispersion of the linear and nonlinear susceptibility can be estimated. This method allows us to investigate also the optical pulses with time duration equal to or shorter than the relaxation time of the media. The electric and magnetic fields are presented as sums of circular and linear components. Thus, NME is written as a set of nonlinear Dirac equations (NDE). Exact solutions of NDE with classical orbital momentum ℓ=1 and opposite directions of the spin (opposite charge) j=±1/2 are obtained. The possible generalization of NME to higher number of optical components and higher number of ℓ and j is discussed. Two kinds of Kerr type media, with and without linear dispersion of the electric and the magnetic susceptibility are considered. The vortex solutions in case of media with dispersion admit finite energy while the solutions in case of a medium without dispersion admit infinite energy.