We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on $\mathbb{Z} ^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+\alpha )}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<\alpha <1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $\alpha \geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.
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