Bounded Time Inverse Scattering for Semilinear Dirac Equation

Abstract

In this paper, we present the first uniqueness result on the bounded time inverse scattering problem for a semilinear Dirac equation with smooth nonlinearity $F(x, z)$ where $(x, z)\in \mathbb{R}^3\times \mathbb{C}^4$ and $x$ is the spatial variable. We show that the solution map, which sends initial data at time 0 to the solution at time $T$, uniquely determines $F(x, z)$ on $x \in \mathbb{R}^3$ and $|z| \leq M$, where $M$ is a constant depend on the solution map, under the assumption that $\partial_z F(x, 0)$ and $\partial^2_z F(x, 0)$ are known. In the proof, we construct a sequence of collisions approaching the initial timeline to simulate a boundary collision. This technique enables us to overcome the difficulties of this hyperbolic system without assumptions on the nonlinearity structure.