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