Inverse problem for wave equation with polynomial nonlinearity

For a wave equation containing nonlinearity in the form of a $n$-th order polynomial, the problem of determining the coefficients of the polynomial depending on the variable $x\in \mathbb{R}^3$ is studied. Plane waves propagating with a sharp front in a homogeneous medium in the direction of a unit vector $\boldsymbol\nu$ and falling on inhomogeneity localized inside some ball $B(R)$ are considered. It is assumed that the solutions of forward problems for all possible $\nu$ can be measured at points of the boundary of this ball at time close to the arrival of the wave front. It is shown that the solution of the inverse problem is reduced to a series of X-ray tomography problems.