An Inverse Problem for the Wave Equation with Two Nonlinear Terms

Abstract

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 \(\mathbf {y}\) 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 \( \mathbf {y}\) 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.