An Inverse Problem for a Semilinear Wave Equation

Abstract

For the equation {{u}_{{tt}}} - Delta u - f(x,u) = 0, (x,t) in {{mathbb{R}}^{4}}, where f(x,u) is a smooth function of its variables and is compact in x, the inverse problem of recovering this function from given information on solutions of Cauchy problems for the differential equation is studied. Plane waves with a strong front that propagate in a homogeneous medium in the direction of the unit vector ν and then impinge on an inhomogeneity localized inside some ball B(R) are considered. It is supposed that the solutions of the Cauchy problems can be measured on the boundary of this ball for all ν at times close to the arriving time of the front. The forward Cauchy problem is studied, and the existence of a unique bounded solution in a neighborhood of a characteristic wedge is stated. An amplitude formula for the derivative of the solution with respect to t on the front of the wave is derived. It is demonstrated that the solution of the inverse problem reduces to a series of X-ray tomography problems.