Crossovers from nonlinear wave-packet acceleration to wave-mixing and self-trapping in the Hatano-Nelson model

Abstract

We demonstrate that wave amplification enables even weak nonlinearities to reshape linear wave-packet transport in nonreciprocal systems. We study the dynamics of bulk Gaussian wave-packets in the Hatano--Nelson model with on-site cubic nonlinearity. We show that the interplay between nonlinearity and amplification generates growing frequency shifts that drive the wave-packet through three successive dynamical regimes: an early nonlinear-skin regime with coherent propagation, an intermediate wave-mixing regime driven by eigenmode resonances, and a self-trapping regime in which part of the packet localizes while the remainder ballistically spreads along the system favored direction. The crossover time scales are set by the width and averaged spacing of the eigenfrequency spectrum. Crucially, within the nonlinear-skin regime, we derive analytical predictions for the wave-packet dynamics and show that nonlinearity couples amplification, dispersion, and nonreciprocity, thereby modifying the magnitude of the wave-packet acceleration and introducing an explicit time dependence into its evolution. Focusing nonlinearities suppress the acceleration and cause it to decrease in time, whereas defocusing nonlinearities enhance it and cause it to increase. We further show that nonlinear interactions typically break down the wave-packet before the non-Hermitian jump can occur. Our results provide a route toward accurate control of waves in nonreciprocal metamaterials.