AN INVERSE PROBLEM FOR ELECTRODYNAMIC EQUATIONS WITH A NONLINEAR CURRENT DEPENDENCE OF A TENSION

The system of Maxwell equations in which a current depends nonlinearly of the electrical tension is considered. In the studying case, it is determined of 4 coefficients depended of space variables. These coefficients are supposed to be finite functions with a support located within ball 𝐵(𝑅) of radius 𝑅. For electrodynamic equations a problem of falling down of a plane running wave with a strong front on the inhomogeneity localized in ball 𝐵(𝑅) is posed. A formula for calculation of an amplitude of this wave is derived. In the sequel, an inverse problem of finding 4 coefficients whose determine the current is considered. For this goal the amplitudes formula for different directions of falling waves is used for points at a part of the boundary of 𝐵(𝑅). It is demonstrated that this inverse problem is decomposed at 4 separated problems. One of them is the usual X-ray tomography problem, when the remain 3 others problems are identical problems of the integral geometry for a family of strait lines. In the latter problems, integrals of an unknown function is given along strait lines with a weight function which depends on the finding coefficients after solving the tomography problem. Arising problems of the integral geometry is studied and stability estimate of its solutions is found.

inverse problem, nonlinear system of Maxwell equations, tomography, integral geometry, stability