An inverse problem for a semi-linear elliptic equation in Riemannian geometries

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 $-{\mathrm{\Delta }}_{g}u+V(x,u)=0$. 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.