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

## TL;DR

This paper investigates the inverse problem of uniquely recovering a complex-valued scalar function on a Riemannian manifold from boundary measurements for a semi-linear elliptic equation, using a constructive approach based on linearization and geodesic transforms.

## Contribution

It introduces a novel method combining multiple linearizations and the Jacobi weighted ray transform to prove uniqueness for a broad class of nonlinearities in inverse problems on Riemannian manifolds.

## Key findings

- Proved uniqueness of the scalar function under certain geometric conditions.
- Developed a constructive proof using complex geometric optics solutions.
- Connected nonlinear interactions to a weighted geodesic transform.

## Abstract

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.