An Inverse Problem for the Wave Equationwith Two Nonlinear Terms

An inverse problem for a second-order hyperbolic equation containing two nonlinear terms is studied. The problem is to reconstruct the coefficients of the nonlinearities. The Cauchy problem with a point source located at a point y is considered. This point is a parameter of the problem and successively runs over a spherical surface S. It is assumed that the desired coefficients are nonzero only in a domain lying inside S. The trace of the solution of the Cauchy problem on S is specified for all possible values of y and for times close to the arrival of the wave from the source to the points on the surface S; this allows reducing the inverse problem under consideration to two successively solved problems of integral geometry. Solution stability estimates are found for these two problems.