  Breathing solitons consist of a fast beating wave within a compact envelope
of stable shape and velocity. They manifest themselves in a variety of contexts
such as plasmas, optical fibers and cold atoms, but have remained elusive when
energy conservation is broken. Here, we report on the observation of breathing,
unidirectional, arbitrarily long-lived solitons in non-reciprocal,
non-conservative active metamaterials. Combining precision desktop experiments,
numerical simulations and perturbation theory on generalizations of the
sine-Gordon and nonlinear Schr\"odinger equations, we demonstrate that
unidirectional breathers generically emerge in weakly nonlinear non-reciprocal
materials, and that their dynamics are governed by an unstable fixed point.
Crucially, breathing solitons can persist for arbitrarily long times provided:
(i) this fixed point displays a bifurcation upon reachin a delicate balance
between energy injection and dissipation; (ii) the initial conditions allow the
dynamics to reach this bifurcation point. Our work establishes non-reciprocity
as a promising avenue to generate stable nonlinear unidirectional waves, and
could be generalized beyond metamaterials to optics, soft matter and
superconducting circuits.
