  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.
