Nonlinear dynamics in PT-symmetric lattices

TL;DR

This paper analyzes nonlinear dynamics in finite PT-symmetric chains, proving boundedness of solutions with small initial data and exponential growth for large data, supported by numerical simulations.

Contribution

It provides rigorous proofs of boundedness and exponential growth in PT-symmetric dNLS chains, extending understanding of stability and instability regimes.

Findings

Solutions with small initial data remain bounded for all times.

Large initial data can lead to exponential growth.

Numerical simulations confirm analytical results.

Abstract

We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schr{\"o}dinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We prove that the solutions of the dNLS equation do not blow up in a finite time and the trajectories starting with small initial data remain bounded for all times. Nevertheless, for arbitrary values of the gain and loss parameter, there exist trajectories starting with large initial data that grow exponentially fast for larger times with a rate that is rigorously identified. Numerical computations illustrate these analytical results for dimers and quadrimers.