A Stability Estimate for a Solution to an Inverse Problem for a Nonlinear Hyperbolic Equation

We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction \( \ell \) falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector \( \ell \) is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem.