Recovery of a general nonlinearity in the semilinear wave equation

We study the inverse problem of recovery a nonlinearity f(t,x,u), which is compactly supported in x, in the semilinear wave equation utt−Δu+f(t,x,u)=0. We probe the medium with either complex or real-valued harmonic waves of wavelength ∼h and amplitude ∼1. They propagate in a regime where the nonlinearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits suppxf. We show that one can recover f(t,x,u) when it is an odd function of u, and we can recover α(x) when f(t,x,u)=α(x)u2m. This is done in an explicit way as h→0.