# Recovery of zeroth order coefficients in non-linear wave equations

## TL;DR

This paper investigates an inverse problem for semi-linear wave equations, demonstrating the unique recovery of a scalar potential function using geometric optics and wave interactions, first in Minkowski space and then in general Lorentzian manifolds.

## Contribution

It introduces a method to recover the potential in semi-linear wave equations on Lorentzian manifolds, extending techniques from Minkowski space to curved spacetimes.

## Key findings

- Proved uniqueness of potential recovery in Minkowski space using wave interactions.
- Generalized the uniqueness result to curved Lorentzian manifolds with Gaussian beams.
- Established a new approach for inverse problems in non-linear wave equations.

## Abstract

This paper is concerned with the resolution of an inverse problem related to the recovery of a scalar (potential) function $V$ from the source to solution map, of the semi-linear equation $(\Box_{g}+V)u+u^3=0$ on a globally hyperbolic Lorentzian manifold $(M,g)$. We first study the simpler model problem where the geometry is the Minkowski space and prove the uniqueness of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.