Modulational Instability and Dynamical Growth Blockade in the Nonlinear Hatano-Nelson Model

TL;DR

This work analyzes modulational instability in the nonlinear Hatano-Nelson model under periodic boundary conditions. The model is governed by $i \frac{d \psi_n}{dt} = \kappa e^{h} \psi_{n+1} + \kappa e^{-h} \psi_{n-1} + \chi |\psi_n|^2 \psi_n$ with $h>0$, and nonlinear plane waves $\psi_n(t)=A e^{i q n - i \omega(q) t - i \theta(t)}$ where $\omega(q)=2 \kappa \cosh(h+ i q)$. The authors show that all plane waves are modulationally unstable for any $q$, including those with $\mathrm{Im}(\omega(q))>0$, and demonstrate a growth blockade: after an initial exponential growth of the total intensity $P(t)=\sum_n|\psi_n|^2$ at rate $2 \kappa \sinh h$, growth halts as the field develops self-induced disorder. A time-averaged potential analysis explains the blockade as a suppression of convective transport by effective disordered potentials, revealing a novel nonlinear-dissipative regime in non-Hermitian lattices with potential applications in optics and mean-field quantum systems.

Abstract

The Hatano-Nelson model is a cornerstone of non-Hermitian physics, describing asymmetric hopping dynamics on a one-dimensional lattice, which gives rise to fascinating phenomena such as directional transport, non-Hermitian topology, and the non-Hermitian skin effect. It has been widely studied in both classical and quantum systems, with applications in condensed matter physics, photonics, and cold atomic gases. Recently, nonlinear extensions of the Hatano-Nelson model have opened a new avenue for exploring the interplay between nonlinearity and non-Hermitian effects. Particularly, in lattices with open boundary conditions, nonlinear skin modes and solitons, localized at the edge or within the bulk of the lattice, have been predicted. In this work, we examine the nonlinear extension of the Hatano-Nelson model with periodic boundary conditions and reveal a novel dynamical phenomenon arising from the modulational instability of nonlinear plane waves: growth blockade. This phenomenon is characterized by the abrupt halt of norm growth, as observed in the linear Hatano-Nelson model, and can be interpreted as a stopping of convective motion arising from self-induced disorder in the lattice.

Figures (5)

• Figure 1: Onset of modulational instability in the DNLSE. (a) Numerically-computed temporal evolution of the amplitudes $|\psi_n(t)|$ on a pseudocolor map in a lattice comprising $N=12$ sites with periodic boundary conditions for hopping rate $\kappa=1$ and nonlinear parameter $\chi=1$. The initial condition is the plane wave $\psi_n(t)=A\exp(iqn)$ with amplitude $A=1$ and spatial wave number $q=0$, perturbed by adding a small random noise. Note that the plane wave is modulationally unstable. (b) Same as (a), but for $\chi=-1$. In this case the modulational instability is absent and the plane wave is stable. (c) Same as (a) but for $q=2 \pi/3$. The plane wave is stable. (d) Same as (c) but for $\chi=-1$. The plane wave is unstable.
• Figure 2: Illustration of the growth blockade effect. (a) Numerically-computed temporal evolution of the amplitudes $|\psi_n(t)|$ on a pseudocolor map in a lattice comprising $N=12$ sites with periodic boundary conditions for parameter values $\kappa=1$ (hopping rate), $h=0.1$ (imaginary gauge field) and for a nonlienar coefficient $\chi=1$ (upper panels) and $\chi=-1$ (lower panels). The initial condition is the plane wave $\psi_n(t)=A\exp(iq_0n)$ with amplitude $A=1$ and spatial wave number $q_0=\pi/2$, perturbed by adding a small random noise. (b) Corresponding temporal evolution of the total excitation intensity $P(t)=\sum_n | \psi_n(t)|^2$ (solid curve). The dashed curve is the behavior of $P(t)$ of the nonlinear plane wave solution, given by the exponential law $P(t)=N|A|^2 \exp \left\{ 2 {\rm Im}(\omega (q_0) ) t \right\}$. Note that the exponential growth of the total excitation intensity is arrested after an initial transient, and $P(t)$ settles down to a nearly constant value with small fluctuations for times $t > \sim 10$. The arrest of the secular intensity growth is associated to the emergence of an irregular spatial distribution of $\psi_n(t)$.
• Figure 3: Same as Fig.2, but for $q_0=0$.
• Figure 4: Same as Fig.2, but for $q_0=-\pi$.
• Figure 5: (a) Numerically-computed behavior of the local averaged potential $\overline{V}_n(t)=(\chi/ \Delta t) \int_t^{t+ \Delta t} |\phi_n(\tau)|^2 d \tau$ versus time $t$, corresponding to the solution $\phi_n(t)$ shown in Fig.2 with $\chi=1$, on a pseudo color map. Average time interval is $\Delta t=1$. (b) Detailed behavior of $\overline{V}_n$ for time $t=19$. (c) Eigenvalue spectrum of the matrix $\mathcal{S}$ with potential $\overline{V}_n$ plotted in panel (b). Note that the imaginary part of all eigenvalues vanishes.