Coexistence of shocks and rarefaction fans: complex phase diagram of a   simple hyperbolic particle system
  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. 
M\'arton Bal\'azs, Attila L\'aszl\'o Nagy, B\'alint T\'oth, Istv\'an   T\'oth
math.PR cond-mat.stat-mech