On the inverse problem for a hyperbolic dispersive partial differential equation

Abstract

The inverse problem for a two-dimensional (space-time) hyperbolic partial differential equation, with coefficients, functions of the spatial variable only, is considered. Exterior to a region of compact support in the spatial variable, the equation reduces to the wave equation, and, from knowledge of the solution in the exterior region (namely in terms of reflected and transmitted waves for a prescribed incident wave), the problem is to deduce the coefficients in the interior region. This is achieved by treating the problem as a Cauchy initial value problem and using the Riemann function to deduce a dual set of integral equations. The coefficients or linear combinations of them are deduced from the solutions of the integral equations. The question of uniqueness is partially answered, by estimating the domain of convergence of the Neumann series. The application of the analysis to electromagnetic scattering from a slab of varying conductivity and permitivity is indicated.