Uniqueness result for semi-linear wave equations with sources

This paper addresses the inverse problem of simultaneously recovering multiple unknown parameters for semilinear wave equations from boundary measurements. We consider an initial-boundary value problem for a wave equation with a general semilinear term and an internal source. The inverse problem is to determine the nonlinear coefficients (potentials), the source term, and the initial data from the Dirichlet-to-Neumann (DtN) map. Our approach combines higher-order linearization and the construction of geometric optics (GO) solutions. The main results establish that while unique recovery is not always possible, we can precisely characterize the gauge equivalence classes in the solutions to this inverse problem. For a wave equation with a polynomial nonlinearity of degree $ n $, we prove that only the highest-order coefficient can be uniquely determined from the DtN map; the lower-order coefficients and the source can only be recovered up to a specific gauge transformation involving a function $ \psi $. Furthermore, we provide sufficient conditions under which unique determination of all parameters is guaranteed. We also extend these results to various specific non-polynomial nonlinearities, demonstrating that the nature of the nonlinearity critically influences whether unique recovery or a gauge symmetry is obtained.