Evolutionary equations for electromagnetic fields in unbounded space filled with layered inhomogeneous nonlinear transient medium with losses

The transformation of Maxwell equations into the set of evolutionary equations is carried out for the case of arbitrary electrodynamic problem in unbounded space filled with layered inhomogeneous nonlinear transient medium with losses. The inhomogeneity of medium is permitted in longitudinal direction only. After elimination of longitudinal components of electromagnetic field the initial problem is converted into two matrix problems. It is proved that the matrix operators are self-adjoint. Eigen functions and eigen numbers of the operators are found. Using the operators to Maxwell equations is the projection of initial equation set into the modal basis in transversal plane. The completeness of the basis is proved by means of Weyl theorem about orthogonal splitting of Hilbert space. As a result, the evolutionary equation set is obtained. The procedure essentially simplifies the solving of the initial nonlinear transient problem because the three-dimensional problem is converted into the problem for one-dimensional evolutionary equations.