One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

For the system of nonlinear electrodynamics equations, we consider the problem of determining the medium conductivity coefficient multiplying the nonlinearity. It is assumed that the permittivity and permeability are constant and the conductivity depends only on one spatial variable, with this conductivity being zero on the half-line x . For a mode in which only two electromagnetic field components participate, the wave propagation process caused by the incidence of a plane wave with a constant amplitude from the domain x<0  onto an inhomogeneity localized on the half-line x<0  is considered. With a given conductivity coefficient, the conditions for the solvability of the direct problem and the properties of its solution are studied. To solve the inverse problem, the trace of the electrical component of the solution of the direct problem is specified on a finite segment of the axis x=0. A theorem on the local existence and uniqueness of the solution of the inverse problem is established, and a global estimate of the conditional stability of its solutions is found.