Nonlinear dynamics in PT-symmetric lattices

We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schrödinger (dNLS) type. For arbitrary values of the gain and loss parameter, we prove that the solutions of the dNLS equation do not blow up in a finite time but nevertheless, there exist trajectories starting with large initial data that grow exponentially fast for larger times with a rate that is rigorously identified. In the range of the gain and loss parameter, where the zero equilibrium state is neutrally stable, we prove that the trajectories starting with small initial data remain bounded for all times. Numerical computations illustrate these analytical results for dimers and quadrimers.