# Speed-of-light pulses in the massless nonlinear Dirac equation with a potential

## TL;DR

This paper investigates the behavior of pulses in the massless nonlinear Dirac equation with various potentials, revealing that initial pulses split into two light-speed solutions while conserving charge and energy, with some cases allowing exact solutions.

## Contribution

The study provides numerical analysis of pulse splitting in the massless nonlinear Dirac equation with external potentials and derives exact solutions for the constant potential case.

## Key findings

- Initial pulses split into two light-speed solutions after a short time.
- Charge and energy are conserved during splitting.
- Exact solutions are derived for the constant potential case.

## Abstract

We consider the massless nonlinear Dirac (NLD) equation in $1+1$ dimension with scalar-scalar self-interaction $\frac{g^2}{2} (\bar{\Psi} \Psi)^2$ in the presence of three external electromagnetic potentials $V(x)$, a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find that, for all three cases, after a short transit time, the initial pulse breaks into two pulses which are solutions of the massless linear Dirac equation traveling in opposite directions with the speed of light. During this splitting the charge and the energy are conserved, whereas the momentum is conserved when the solutions possess specific symmetries. For the case of the constant potential, we derive exact analytical solutions of the massless NLD equation that are also solutions of the massless linearized Dirac equation.