Modulation instability in dispersive parity-broken systems

Abstract: 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.