Recovery of a Null Form in the Wave Equation from Scattering Data

We use highly oscillatory geometric optics solutions to solve the inverse problem for the system $$ \square \begin{bmatrix} u^{(1)}\\ u^{(2)}\\ \vdots\\ u^{(n)}\end{bmatrix} = \sum_{\substack{k,l=1\\k\geq l}}^n \left(Q_0(u^{(k)},u^{(l)}) \begin{bmatrix} q_{1kl}\\q_{2kl}\\\vdots\\q_{nkl} \end{bma