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 - Δ u + α ( x ) | u | 2 u = 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 supp α . We show that one can extract the Radon transform of α from the phase shift of such waves, and then one can recover α . We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem.