  This work explores the interplay between dispersive parity breaking and
non-linearity in two contrasting continuous dynamical systems that exhibit
Modulation Instability (MI). We begin by examining deep water odd surface
gravity waves and derive the non-linear Schr\"odinger equation (NLSE) for the
modulated envelope dynamics using the method of multiple scales. The parity
breaking in the odd gravity waves results in distinct NLSEs for the right and
the left mover, leading to chirality-dependent stability properties for the
envelope dynamics. Moreover, the resonant interaction of gravity waves and odd
viscosity-induced capillary dynamics creates a window of wave numbers in one of
the chiral sectors where the envelope propagation remains stable. Following the
odd gravity results, we design a one-dimensional non-reciprocal PT-symmetric
dielectric model that exhibits parity-breaking effects that are analogous to
the odd viscosity term in 2D hydrodynamics. With cubic non-linearity in the
polarization dynamics, we derive the corresponding NLSE. Once again, we observe
that parity breaking stabilizes the modulated envelope dynamics in the lower
polariton bands. We then compare the similarities and differences between this
case and that of odd gravity waves.
