Bounded Time Inverse Scattering for Semilinear Dirac Equation

TL;DR

This paper proves the first uniqueness result for bounded time inverse scattering of a semilinear Dirac equation, showing that the solution map uniquely determines the nonlinearity under certain conditions.

Contribution

It introduces a novel approach using collision sequences to establish uniqueness without assumptions on the nonlinearity structure.

Findings

Unique determination of the nonlinearity from the solution map

Construction of collision sequences to simulate boundary interactions

Overcoming hyperbolic system difficulties without structural assumptions

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.