Dark states and vortex solitary wave with finite energy of Maxwell-Dirac nonlinear equations

Starting from the Maxwell’s equations for media with no-stationary linear and nonlinear polarization, we obtain a set of nonlinear Maxwell amplitude equations in the approximation of the first order of dispersion. After a special kind of complex presentation, the set of amplitude equations is written as a set of nonlinear Dirac equations. For broad-band pulses solitary solutions with half-integer spin and finite energy are found. The solutions correspond to an electromagnetic wave with circular Poynting vector and zero divergence. These invisible for the detectors waves are called dark states and the localized energy is determinates as electromagnetic mass.