  In recent years, nonreciprocally coupled systems have received growing
attention. Previous work has shown that the interplay of nonreciprocal coupling
and Goldstone modes can drive the emergence of temporal order such as traveling
waves. We show that these phenomena are generically found in a broad class of
pattern-forming systems, including mass-conserving reaction--diffusion systems
and viscoelastic active gels. All these systems share a characteristic
dispersion relation that acquires a non-zero imaginary part at the edge of the
band of unstable modes and exhibit a regime of propagating structures
(traveling wave bands or droplets). We show that models for these systems can
be mapped to a common "normal form" that can be seen as a spatially extended
generalization of the FitzHugh--Nagumo model, providing a unifying
dynamical-systems perspective. We show that the minimal nonreciprocal
Cahn--Hilliard (NRCH) equations exhibit a surprisingly rich set of behaviors,
including interrupted coarsening of traveling waves without selection of a
preferred wavelength and transversal undulations of wave fronts in two
dimensions. We show that the emergence of traveling waves and their speed are
precisely predicted from the local dispersion relation at interfaces far away
from the homogeneous steady state. Our work thus generalizes previously studied
nonreciprocal phase transitions and shows that interfaces are the relevant
collective excitations governing the rich dynamical patterns of conserved
fields.
