Stability analysis for solitons in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric optical lattices

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.

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