Evolutionary approach for the problems of transient electromagnetic field propagation in nonlinear medium

The problem of propagation of transient electromagnetic field in unbounded nonlinear medium is converted to the set of evolutionary equations in time domain by the modal representation of the field. Using evolutionary approach the initial nonstationary three-dimensional electrodynamic problem is transformed to the problem for one-dimensional evolutionary equation set. The modal basis for electromagnetic fields with arbitrary time dependence in cylindrical coordinate system is used. After elimination of the longitudinal components of electrical and magnetic field from Maxwell´s equation system the four-dimensional differential operators are formed. It is proved that the operators are self-adjoint ones with energy measure. The eigen-functions of the operators form the basis in transversal plane. The completeness of the basis is proved by means of Weyl Theorem about orthogonal splitting of the four dimensional Hilbert space. The expansion coefficients of transient electromagnetic field can be found from the set of evolutionary equations obtained.