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 $|z|\le M$, where $M$ is a constant depend on the solution map, under the assumption that ${\partial }_{z}F(x,0)$ and ${\partial }_z^{2}F(x,0)$ 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.

22 pages, 2 figures