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

TL;DR

This paper explores how wave amplification and nonlinearities in the Hatano-Nelson model lead to complex wave dynamics, including regimes of propagation, wave-mixing, and self-trapping, with implications for controlling waves in nonreciprocal systems.

Contribution

It reveals the interplay between nonlinearity, amplification, and nonreciprocity, providing analytical predictions and identifying regimes of wave behavior in the Hatano-Nelson model.

Findings

Nonlinearity and amplification generate frequency shifts driving different dynamical regimes.

Analytical predictions describe wave-packet dynamics in the nonlinear-skin regime.

Nonlinear interactions can prevent the non-Hermitian jump by breaking down the wave packet.

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 onsite 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 mode 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 average spacing of the eigen-frequency 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.