Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime

We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation $u_{tt}-Delta u+ alpha(x) |u|^2u=0$, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength $h$ and amplitude $h^{-1/2}$, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits the support of $alpha$. We show that one can extract the Radon transform of $alpha$ from the phase shift of such waves, and then one can recover $alpha$. We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem.