Stability of solitons in parity-time ($\mathcal{PT}$)-symmetric periodic potentials (optical lattices) is analyzed in both one- and two-dimensional systems. First we show analytically that when the strength of the gain-loss component in the $\mathcal{PT}$ lattice rises above a certain threshold (phase transition point), an infinite number of linear Bloch bands turn complex simultaneously. Second, we show that while stable families of solitons can exist in $\mathcal{PT}$ lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. Specifically, when the gain-loss component increases, the parameter range of stable solitons shrinks as new regions of instability appear. Third, we investigate the nonlinear evolution of unstable $\mathcal{PT}$ solitons under perturbations, and show that the energy of perturbed solitons can grow unbounded even though the $\mathcal{PT}$ lattice is below the phase transition point.

• Received 12 January 2012

DOI:https://doi.org/10.1103/PhysRevA.85.023822