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 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.

Figures (8)

• Figure 1: (a) Schematic of the nonlinear HN chain. (b) Effect of nonreciprocity and dispersion on an initial wave packet centered at $k_0=0$ in the mode space. (c) Same as (b), but illustrating the effect of nonlinearity. (d) Theoretical dynamical regimes of nonlinear wave packets in the HN model: (NS) nonlinear-skin, (WM) wave-mixing, and (ST) self-trapping regimes. The parameter space shows the nonlinear coefficient $\alpha$ against time $t$ with the boundaries, $T_{\mathrm{mix}}$ and $T_{\mathrm{trap}}$ obtained comparing the nonlinear frequency shift $\nu(t)$ to the linear eigen-frequency scales, $\Delta$ and $d$. The lower boundary $T_{\mathrm{mix}}$ is calculated from $\nu(T_{\mathrm{mix}})=d$, while the upper one is determined by $\nu(T_{\mathrm{trap}})=\Delta$ (see text for details).
• Figure 2: Analytical estimates. (a) Schematic of the potential $U_\sigma$ associated with the Hamiltonian formulation of the equation governing the nonlinear wave-packet width in the focusing case, $\alpha>0$. Its single fixed point decreases in time, and the surrounding orbits $\sigma(t)$ follow the same trend. (b) Phase portrait of the width dynamics in the plane $(\sigma,\dot{\sigma})$. The gray arrows indicate the direction of time for $\alpha<0$ and $\alpha\geq0$. (c) Analytical predictions for the acceleration $\ddot{X}$ obtained from the collective-coordinate approach [Eq. \ref{['eq:ansatz_nonlinear']}]. The dark red and blue shaded regions indicate the trends associated with focusing and defocusing nonlinearities, respectively.
• Figure 3: Numerical results. Dependence of the participation number $P$ on time $t$ for an initial Gaussian wave packet with $\sigma_0=8$, $X_0 = 0$, and $k_0 = 0$ in a HN lattice with $M=512$ sites and $g=1.25$. The curves correspond to $\alpha=0$ (black), $\alpha=0.0125$ (blue), $\alpha=0.1$ (red), $\alpha=0.8$ (cyan), and $\alpha=100$ (magenta). Star symbols of the same color as the curve above indicate the time $T_{\mathrm{mix}}$ at which $P(t)$ departs from a linear-like behavior (black curve). Similarly, filled dot symbols indicate the time $T_{\mathrm{trap}}$ at which $P(t)$ reaches its minimum. The dotted line guides the eye to the ballistic spreading observed in the self-trapping regime. After passing, $T_{\mathrm{trap}}$ the $P(t)$ in all cases follow this ballistic growth, $P \sim t$ or $\sigma^2 \sim t^2$ (see black dashed line).
• Figure 4: Numerical results. Spatiotemporal evolution of the renormalized amplitude distribution $\lvert y_n(t)\rvert^2$ for a lattice with $M=512$ sites and OBCs at $g=1.25$, initialized with a Gaussian wave packet with $\sigma_0=8$, $X_0=0$, and $k_0=0$, see Fig. \ref{['fig:participation_ratio_nonrecip_weak_all']}. (a)–(d) We plot four representative cases with (a) $\alpha=0$, (b) $\alpha=0.01$, (c) $\alpha=0.8$, and (d) $\alpha=100$. In all panels, points are colored according to the magnitude of the wave-packet amplitude (see top-colorbar). The red curve shows the COM trajectory [Eq. \ref{['eq:center_of_mass_numerical']}]. In addition, the green dashed and dotted horizontal lines mark the characteristic times $T_{\mathrm{mix}}$ and $T_{\mathrm{trap}}$, which separate the nonlinear-skin, wave-mixing, and self-trapping regimes. Panels (e) and (f) show snapshots of the spatial profiles at times $t=20$ (blue), $t=95$ (red), $t=112$ (cyan), and $t=150$ (magenta) for (e) $\alpha=0$ and (f) $\alpha=0.01$. In both panels, the black dashed arrows indicate wave amplification.
• Figure 5: Numerical results. Spatio-temporal evolution of the renormalized amplitude distribution $|c_k(t)|^2$ in the mode variables, $c_k$ with $y_n = \sum_{k} c_k u_{k,n}$ (see text details) using the same sets of parameters as in Fig. \ref{['fig:heatmaps_real_space_01a']}. Panels (a)–(d) correspond to four representative cases with (a) $\alpha=0$, (b) $\alpha=0.01$, (c) $\alpha=0.8$, and (d) $\alpha=100$. The color codes indicate the magnitude of the wave-packet amplitude (see top-colorbar). We only depicts $k\in [-\pi/2, \pi/2]$ for clarity. The red curve shows the COM trajectory [Eq. \ref{['eq:center_of_mass_numerical']}]. The green dashed and dotted horizontal lines mark the characteristic times $T_{\mathrm{mix}}$ and $T_{\mathrm{trap}}$, see also Figs. \ref{['fig:heatmaps_real_space_01a']}(a)-(d). Panels (e) and (f) show snapshots of the modal profiles at times $t=20$ (blue), $t=95$ (red), $t=112$ (cyan), and $t=150$ (magenta) for (e) $\alpha=0$ and (f) $\alpha=0.01$. In both panels, black dashed arrows indicate mode amplification.