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.

中文翻译：

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黎曼几何中半线性椭圆方程的一个反问题

我们研究了复值标量函数 $V:\mathcal M \times \mathbb C\to \mathbb C$ 的唯一恢复的逆问题，定义在光滑紧凑的黎曼流形 $(\mathcal M,g)$ 上平滑边界，给定 Dirichlet 到 Neumann 映射，在合适的意义上，对于椭圆半线性方程 $-\Delta_{g}u+V(x,u)=0$。我们表明，在一些几何假设下，可以证明一大类非线性的唯一性。该证明是建设性的，并且基于线性化算子的复杂几何光学解附近的半线性方程的多重线性化以及由此产生的非线性相互作用。这些非线性相互作用导致了对沿测地线的加权变换的研究，我们称之为 Jacobi 加权射线变换。