An Inverse Problem for Electrodynamic Equationswith a Nonlinear Dependence of the Current Density on the Electric Field

We consider the system of Maxwell equations in which the current density depends nonlinearly on the electric field. In our case, the system is determined by four coefficients depending on the spatial variables. These coefficients are supposed to be compactly supported in a ball $$B(R) $$ of radius $$R $$ . For the electrodynamic equations, we pose a problem on a plane running wave with a sharp front incident on an inhomogeneity localized in the ball $$B(R) $$ . A formula for the calculation of the wave amplitude is derived. Further, we consider an inverse problem of finding the four coefficients determining the current from the wave front amplitude given for various directions of the incident plane wave on part of the boundary of $$B(R)$$ . We show that this inverse problem splits into four separate problems, one of which is the usual X-ray tomography problem and the other three are identical problems of integral geometry for a family of straight lines. These problems are studied, and a stability estimate for their solutions is found.