  Stability of solitons in parity-time (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
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 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. Thirdly, we investigate the nonlinear evolution of unstable PT solitons
under perturbations, and show that the energy of perturbed solitons can grow
unbounded even though the PT lattice is below the phase transition point.
