Abstract
A quasilinear hyperbolic equation is considered whose principal part is a purely wave operator and the lower order part consists of two nonlinear terms with coefficients \(p\) and \(q\) compactly supported in a ball \(B\) . We study the direct problem of a plane wave scattered by a heterogeneity localized in \(B\) and the inverse problem of recovering the coefficients \(p\) and \(q\) from solutions of direct problems with a varying incident wave direction. An asymptotic expansion of the solution to the direct problem near the front of the traveling plane wave is presented, based on which the inverse problem is reduced to two linear problems to be solved sequentially. Namely, the problem of determining the coefficient \(p\) is reduced to a classical X-ray tomography problem, while the problem of determining the coefficient \(q\) is reduced to a more complicated problem of integral geometry. The last problem, which is new, is to find a function from its integrals with a given weight along straight lines. This problem is investigated, and a uniqueness and stability theorem for its solution is proved.
