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

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