An inverse problem for a semilinear wave equation

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 $\nu$ 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 $\nu$ 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.