  We study the inverse problem of unique recovery of a complex-valued scalar
function $V:\mathcal M \times \mathbb C\to \mathbb C$, defined over a smooth
compact Riemannian manifold $(\mathcal M,g)$ with smooth boundary, given the
Dirichlet to Neumann map, in a suitable sense, for the elliptic semi-linear
equation $-\Delta_{g}u+V(x,u)=0$. We show that under some geometrical
assumptions uniqueness can be proved for a large class of non-linearities. The
proof is constructive and is based on a multiple-fold linearization of the
semi-linear equation near complex geometric optic solutions for the linearized
operator and the resulting non-linear interactions. These non-linear
interactions result in the study of a weighted transform along geodesics, that
we call the Jacobi weighted ray transform.
