Strong Hydrodynamic Limit for Attractive Particle Systems on $\mathbb{Z}$

We prove almost sure Euler hydrodynamics for a large class of attractive particle systems on Z starting from an arbitrary initial profile. We generalize earlier works by Seppalainen ( 1999) and Andjel et al. ( 2004). Our constructive approach requires new ideas since the subadditive ergodic theorem ( central to previous works) is no longer effective in our setting.