Inverse Problem for the Wave Equation with a Polynomial Nonlinearity

Abstract— For the wave equation containing a nonlinearity in the form of an \( n \) th order polynomial, we study the problem of determining the coefficients of the polynomial depending on the variable \( x\in \mathbb {R}^3 \) . We consider plane waves that propagate in a homogeneous medium in the direction of a unit vector \( \boldsymbol \nu \) with a sharp front and incident on an inhomogeneity localized inside a certain ball \( B(R) \) . It is assumed that the solutions of the problems can be measured at the points of the boundary of this ball at the instants of time close to the arrival of the wavefront for all possible values of the vector \( \boldsymbol \nu \) . It is shown that the solution of the inverse problem is reduced to a series of X-ray tomography problems.