On integrable nonlinear Dirac equations with 1+3 dimension

In this paper, we construct an exactly integrable relativistic model of a system of four fermions in a proper field with a vector potential with values in the matrix algebra 4×4, which are described by the nonlinear Dirac equationin 1+3 dimensions. The proposed approach is based on the method of multidimensional matrix substitutions. A general scheme for constructing exact solutions of the nonlinear Dirac equation and its perturbations near the exact solution is considered. The general aspects of the constructed model and its connection with Yang-Mills theory are discussed.