Lattice solitons in optical media described by the complex Ginzburg-Landau model with $\mathcal{PT}$-symmetric periodic potentials

We report the existence, stability, and rich dynamics of dissipative lattice solitons in optical media described by the cubic-quintic complex Ginzburg-Landau model with parity-time ($\mathcal{PT}$) symmetric potentials. We focus on studying the generic spatial soliton propagation scenarios by changing (a) the linear loss coefficient in the complex Ginzburg-Landau model, (b) the amplitudes, and (c) the periods of real and imaginary parts of the complex-valued $\mathcal{PT}$-symmetric optical lattice potential. Generically, it is found that if the period of the real part of the $\mathcal{PT}$-symmetric optical lattice potential is close to $\pi ,$ the spatial solitons are tightly bound and they can propagate straightly along the lattice, while if the period of the real part of the $\mathcal{PT}$-symmetric optical lattice potential is larger than $\pi ,$ the launched solitons are loosely bound and they can exhibit either a transverse (lateral) drift or a persistent swing around the input launching point due to gradient force arising from the spatially inhomogeneous loss. These latter features are intimately related to the dissipative nature of the system under consideration because they do not arise in the conservative counterpart of the dynamical model. These generic propagation scenarios can be effectively managed by properly changing the profile of the spatially inhomogeneous loss.

Figure 1

Soliton dynamics for the case of a tight-binding lattice potential with a relatively small period $T_1$ $=$ 1 of the real part of the $\mathcal{PT}$-symmetric potential. (a) The dependence of the linear loss coefficient α on the period $T_2$ of the imaginary part of the $\mathcal{PT}$-symmetric OL periodic potential indicating the domains of different propagation scenarios: excess gain propagation (region A), soliton drift to adjacent lattice (region B), stable straight propagation (region C), and soliton decay (for $\alpha >0.54$). Evolution scenarios of the lattice solitons: (b) excess gain propagation for $\alpha =0.2$ and $T_2$ $=$ 0.55, (c) soliton drift for $\alpha =0.25$ and $T_2$ $=$ 0.55, (d) soliton drift for $\alpha =0.4$ and $T_2$ $=$ 0.55, (e) straight propagation for $\alpha =0.5$ and $T_2$ $=$ 0.55, (f) soliton decay for $\alpha =0.55$ and $T_2$ $=$ 0.55. Other parameters are $A_1$ $=$ 0.2 and $A_2$ $=$ 0.2.

Figure 2

Soliton dynamics for the case of a large lattice period $T_1$ > 1 of the real part of the $\mathcal{PT}$-symmetric OL potential. (a) The dependence of the linear loss coefficient α on the period $T_1$ of the real part of the $\mathcal{PT}$-symmetric OL potential indicating the regions of different propagation scenarios: soliton excess gain propagation (region A), soliton drift (region B), soliton persistent swing (region C), and soliton decay (for $\alpha >0.54$). Soliton evolution scenarios: (b) soliton excess gain propagation for $\alpha =0.2$ and $T_1$ $=$ 4, (c) soliton drift for $\alpha =0.3$ and $T_1$ $=$ 4, (d) soliton persistent swing for $\alpha =0.4$ and $T_1$ $=$ 4, (e) soliton persistent swing for $\alpha =0.5$ and $T_1$ $=$ 4, (f) soliton decay for $\alpha =0.55$ and $T_1$ $=$ 4. Other parameters are $T_2$ $=$ 0.5, $A_1$ $=$ 0.2, and $A_2$ $=$ 0.2.

Figure 3

Soliton dynamics for the case of a large lattice period $T_2$ of the imaginary part of the $\mathcal{PT}$-symmetric OL potential (a) The dependence of the linear loss coefficient $\alpha$ on the period $T_1$ of the real part of the $\mathcal{PT}$-symmetric OL potential indicating the regions of different propagation scenarios: soliton excess gain propagation (region A), soliton drift (region B), soliton persistent swing (region C), and soliton decay (for $\alpha >0.54$). Soliton evolution scenarios: (b) soliton excess gain propagation for $\alpha =0.2$ and $T_1$ $=$ 2, (c) drift for $\alpha =0.25$ and $T_1$ $=$ 2, (d) soliton persistent swing for $\alpha =0.4$ and $T_1$ $=$ 2, (e) soliton drift for $\alpha =0.25$ and $T_1$ $=$ 6, (f) soliton persistent swing for $\alpha =0.4$ and $T_1$ $=$ 6. Other parameters are $T_2$ $=$ 5, $A_1$ $=$ 0.2, and $A_2$ $=$ 0.2.

Figure 4

Effect of varying the amplitude $A_1$ of the real part of the $\mathcal{PT}$-symmetric OL potential on the persistent swing evolution scenario. (a) $A_1$ $=$ 0.4, (b) $A_1$ $=$ 1.0. Other parameters are $T_1$ $=$ 10, $T_2$ $=$ 5, $A_2$ $=$ 0.2, and $\alpha =0.5$.

Figure 5

Effect of varying the amplitude $A_2$ of the imaginary part of the $\mathcal{PT}$-symmetric OL potential on the persistent swing evolution scenario. (a) $A_2$ $=$ 0.3, (b) $A_2$ $=$ 0.2. Other parameters are $T_1$ $=$ 10, $T_2$ $=$ 5, $A_1$ $=$ 1, and $\alpha =0.4$.