  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.
