We obtain optical vortices with classical orbital momentum
ℓ=1 and spin j=±1/2
as exact solutions of a system of
nonlinear Maxwell equations (NMEs). Two kinds of Kerr-type media,
namely, those with and without linear dispersion of the electric
and the magnet susceptibility, are investigated. The electric and
magnetic fields are represented as sums of circular and linear
components. This allows us to reduce the NME to a set of
nonlinear Dirac equations (NDEs). The vortex solutions in the case
of media with dispersion admit finite energy, while the
solutions in case of media without dispersion admit infinite
energy. The amplitude equations are obtained from equations of
nonstationary optical and magnetic response (dispersion). This
includes also the optical pulses with time duration of order of
and less than the time of relaxation of the media (femtosecond
pulses). The possible generalization of NME to a higher number of
optical components and a higher number of ℓ and j is discussed.
