  Pair interactions between active particles need not follow Newton's third
law. In this work we propose a continuum model of pattern formation due to
non-reciprocal interaction between multiple species of scalar active matter.
The classical Cahn-Hilliard model is minimally modified by supplementing the
equilibrium Ginzburg-Landau dynamics with particle number conserving currents
which cannot be derived from a free energy, reflecting the microscopic
departure from action-reaction symmetry. The strength of the asymmetry in the
interaction determines whether the steady state exhibits a macroscopic phase
separation or a traveling density wave displaying global polar order. The
latter structure, which is equivalent to an active self-propelled smectic
phase, coarsens via annihilation of defects, whereas the former structure
undergoes Ostwald ripening. The emergence of traveling density waves, which is
a clear signature of broken time-reversal symmetry in this active system, is a
generic feature of any multi-component mixture with microscopic non-reciprocal
interactions.
