Nonlinear Amplitude Maxwell–Dirac Equations. Optical Leptons

We apply the method of slowly-varying amplitudes of the electrical and magnetic fields to integro-differential system of nonlinear Maxwell equations. The equations are reduced to system of differential Nonlinear Maxwell amplitude Equations (NME). 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 = 1 and opposite directions of the spin (opposite charge) j = ±1/2 are obtained. Using the Poynting vector for solutions with spin j = 1/2 we find that the energy flow through arbitrary closed surface around our vortex solutions is zero and the localized energy of our solutions circulate in x, y plane. Other important result is that the vortex solutions with spin j = 1/2 without external fields are immovable. The initial investigations on stability of these solutions show that vortices with spin j = 1/2 are stable while the vortices with opposite spin (charge) j = −1/2 are not. The possible generalization of NME to higher number of optical components and higher number of and j is discussed.