This paper is concerned with the resolution of an inverse problem related to the recovery of a 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 $({\mathcal{M}},g)$. We first study the simpler model problem, where $({\mathcal{M}},g)$ is the Minkowski space, and prove the unique recovery 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.

中文翻译：

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非线性波方程中零阶系数的恢复

本文关注与半线性方程$(\Box _{g}+V)u+u^{3从源到解映射的函数$V$恢复相关的逆问题的求解}=0$在全局双曲洛伦兹流形$({\mathcal{M}},g)$上。我们首先研究更简单的模型问题，其中$({\mathcal{M}},g)$是 Minkowski 空间，并通过使用几何光学和产生的三重波相互作用证明了$V$的唯一恢复从三次非线性。随后，使用高斯光束将结果推广到全局双曲洛伦兹流形。