  This paper investigates the non-equilibrium hydrodynamic behavior of a simple
totally asymmetric interacting particle system of particles, antiparticles and
holes on $\mathbb{Z}$. Rigorous hydrodynamic results apply to our model with a
hydrodynamic flux that is exactly calculated and shown to change convexity in
some region of the model parameters. We then characterize the entropy solutions
of the hydrodynamic equation with step initial condition in this scenario which
include various mixtures of rarefaction fans and shock waves. We highlight how
the phase diagram of the model changes by varying the model parameters.
