Staggered parity-time-symmetric ladders with cubic nonlinearity

We introduce a ladder-shaped chain with each rung carrying a parity-time- $\left(\mathcal{PT}\text{−}\right)$ symmetric gain-loss dimer. The polarity of the dimers is staggered along the chain, meaning alternation of gain-loss and loss-gain rungs. This structure, which can be implemented as an optical waveguide array, is the simplest one which renders the system $\mathcal{PT}$-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schrödinger equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anticontinuum limit of uncoupled rungs and using the Newton's method for continuation of the solutions with the increase of the inter-rung coupling, we construct families of $\mathcal{PT}$-symmetric discrete solitons and identify their stability regions. Waveforms stemming from a single excited rung and double ones are identified. Dynamics of unstable solitons is investigated too.

Figure 1

The staggered ladder-shaped lattice with horizontal $\left(C\right)$ and vertical $\left(\kappa \right)$ coupling constants. Red and blue dots designate sites carrying the mutually balanced gain and loss, respectively. The dashed lines designate the horizontal and vertical axes of the $\mathcal{PT}$ symmetry.

Figure 2

Plots of ${log}_{10}\left(\right|u_n{|}^2)$, where $u_n$ at $C=0$ is given by Eqs. (11) and (12), and at $C>0$ the soliton solutions $u_n$ are obtained by the continuation in $C$ (see the text). Common parameters are $\gamma =1,\kappa =1.9$, and $\sigma =1$. The initial configuration of the excited rungs at $C=0$ and parameters are a single in-phase rung with ${\delta }_1={\delta }_{\mathrm{in}},\Lambda =2.5,N=40,\Delta C=0.001$ (top left), a single out-of-phase rung with ${\delta }_1={\delta }_{\mathrm{out}},\Lambda =2.5,N=40,\Delta C=0.001$ (top right), two in-phase rungs with ${\delta }_1={\delta }_2={\delta }_{\mathrm{in}},\Lambda =2,N=80,\Delta C=0.001$ (middle left), two out-of-phase rungs with ${\delta }_1={\delta }_2={\delta }_{\mathrm{out}},\Lambda =2.5,N=80,\Delta C=0.001$ (middle right), and, finally, a mixed state carried by two rungs with ${\delta }_1={\delta }_{\mathrm{in}},{\delta }_2={\delta }_{\mathrm{out}},\Lambda =2.5,N=80,\Delta C=0.00001$ (bottom center). Plots of ${log}_{10}\left(\right|v_n{|}^2)$ are identical to those of ${log}_{10}\left(\right|u_n{|}^2)$. As $C$ increases, small amplitudes appear at adjacent rungs, and the soliton gains width. The corresponding second moment $w\left(C\right)$, defined as per Eq. (18), is shown in Fig. 4. The stability of the solitons shown here is predicted by eigenvalue plots displayed in Fig. 12 for $\gamma =1$.

Figure 3

Profiles of discrete solitons for $C=0.4$. The configurations of the initial $\left(C=0\right)$ solution and other parameters follow the same pattern as in Fig. 2.

Figure 4

Width diagnostic $w$, defined as per Eq. (18), corresponding to each of the plots in Fig. 2.

Figure 5

The phase shift between two edges of the rungs $\left|\arg \right(v_n{u}_n^{*}\left)\right|$, plotted as a function of $C$ and $n$, where $u$ is the solution whose absolute value is presented in Fig. 2. The value of $\left|\arg \right(v_n{u}_n^{*}\left)\right|$ is set to zero for any $n$ at which ${log}_{10}{\left(\right|u|}^2)\le -6$ in Fig. 2. In other words, the phase shift is shown as equal to zero when the amplitude is too small. For the top left and middle left plots, the soliton's field is different from zero at one or two in-phase rung(s) when $C=0$, and as $C$ increases the solutions stay in phase. For the top right and middle right plots, the field at $C=0$ is nonzero and out of phase at the one or two central rungs, and, as $C$ increases, the fields at these rungs, and at two rungs on either side of the central ones, tend to be out of phase, while the field at other rungs, located farther away, tend to be in phase. Similarly, in the bottom plot, where at $C=0$ the $n=1$ rung is in phase and the $n=2$ one is out of phase, as $C$ increases, most rungs tend to be in phase, except for $n=2,3$.

Figure 6

The phase of each rung ${\theta }_u:=\arg \left(u_n\right)$ (blue lines) and ${\theta }_v:=\arg \left(v_n\right)$ (green lines) with values in the interval $(-\pi ,\pi ]$, plotted as a function of the spatial variable $n$. The top four plots and bottom-most left plot have fixed parameter values $C=0.4,\gamma =1$, and $\sigma =1$. That is, the $u_n,v_n$ solutions for these five plots are the same as those whose modulus is plotted in Fig. 3. In the bottom right plot, we show ${\theta }_u,{\theta }_v$ for the same fixed parameter values as the bottom left plot except with $\gamma =0.8$. The bottom two plots show the different phase profiles that can arise for different $\gamma$ values, with either same (left) or opposite (right) signs for the phase on the outer portions of the ladder.

Figure 7

The same as Fig. 2, but for $\sigma =-1$. Common parameters are $\kappa =2$ and $\gamma =1$. The initial values of ${\delta }_1$ and ${\delta }_2$ at $C=0$ follow the pattern of Fig. 2. Other parameters are $\Lambda =5,N=80,\Delta C=0.001$ on the top left, $\Lambda =3.5,N=80,\Delta C=0.001$ on the top right, $\Lambda =5,N=80,\Delta C=0.001$ on the middle left, $\Lambda =3.5,N=40,\Delta C=0.001$ on the middle right, and finally $\Lambda =3.085,N=40,\Delta C=0.0001$ on the bottom center. As $C$ increases, small amplitudes appear at adjacent sites, and the soliton gains width, as shown by means of $w$ in Fig. 9. The stability of the solitons shown here is predicted by the eigenvalue plots in Fig. 13 at $\gamma =1$.

Figure 8

Profiles of the discrete solitons at $C=0.4$, for $\sigma =-1$. The configurations of the initial $C=0$ solution and parameters follow the same pattern as in Fig. 7.

Figure 9

The width diagnostic of the discrete solitons, defined as per Eq. (18), corresponding to each of the plots in Fig. 7.

Figure 10

The same as Fig. 5 but for $\sigma =-1$, i.e., for the discrete solitons presented in Fig. 7. For the top left and middle left plots, the soliton's field is nonzero at one or two in-phase rung(s) when $C=0$, and as $C$ increases these rungs remain in phase, while all the others are out of phase. For the top right and middle right plots, the soliton's field at $C=0$ is nonzero and out of phase at one or two central rungs, and all rungs remain out of phase with the increase of $C$. In the bottom plot, only the $n=1$ rung remains in phase, while all others are out of phase.

Figure 11

The phase of each rung ${\theta }_u:=\arg \left(u_n\right)$ (blue lines) and ${\theta }_v:=\arg \left(v_n\right)$ (green lines) in the interval $(-\pi ,\pi ]$, plotted as a function of the spatial variable $n$ for the fixed parameter values $C=0.4,\gamma =1$, and $\sigma =-1$. That is, the $u_n,v_n$ solutions are the same as those whole modulus is plotted in Fig. 8.

Figure 12

The largest instability growth rate $max\left[\mathrm{Re}\right(i\omega \left)\right]$, determined by matrix $M$ in Eq. (5), for parameter values following the same pattern as in Fig. 2, except that here $\gamma$ varies along the horizontal axis. In the top two plots in the right column, the cyan line represents the analytically predicted critical value ${\gamma }_{\mathrm{cr}}^{\left(1\right)}\left(C\right)=\kappa -C$; this line originates from the cyan dot in the corner at ${\gamma }_{\mathrm{cr}}^{\left(1\right)}\left(0\right)=\kappa$ [see Eq. (9)]. If an in-phase excited rung is present, then the second cyan dot is located at ${\gamma }_{\mathrm{cr}}^{\left(2\right)}=\sqrt{{\kappa }^2-{\Lambda }^2/4}$, in accordance with Eq. (16). Green lines indicate stability boundaries, between the dark region corresponding to stability {or very weak instability, with $max\left[\mathrm{Re}\left(i\omega \right)\right]<{10}^{-3}$}, and the bright region corresponding to the instability.

Figure 13

The same as in Fig. 12, but for parameter values from Fig. 7. Here, the cyan line is drawn on the top two plots in the left column, and the second cyan dot on the $C=0$ axis appears only in the case where the out-of-phase excited rung is present at $C=0$.

Figure 14

Stability eigenvalues $i\omega$ in the complex plane, for parameters chosen in accordance with the topmost left panel of Fig. 12 with $\gamma =0.5$. For $C=0$, in the top left plot we show the agreement of the numerically found eigenvalues (blue circles) with results produced by Eqs. (14) (green filled circles) and (15) (red filled). For $C=0.3$, in the top right panel we show that the eigenvalues associated with the zero solution indeed lie within the predicted intervals (10), the boundaries of which are shown by dashed lines. Next, for $C=1$, in the bottom left plot we observe that values of $i\omega$ associated with the excited state have previously (at smaller $C$) merged with the dashed intervals, and now an unstable quartet has emerged from the axis. For $C=1.5$, in the bottom right panel the critical point corresponding to Eq. (9) is represented, where unstable eigenvalues emerge from the axis at the values of $\pm (\Lambda +C)$, as the intervals in Eq. (10) merge. Comparing plots in the bottom row, we conclude that the critical point of the latter type gives rise, in general, to a stronger instability than the former one.

Figure 15

The same as in Fig. 14, but for parameters chosen in accordance with the top right panel of Fig. 13, with $\gamma =1.2$. At $C=0$, in the top left plot we show the agreement of the numerically found eigenvalues (blue circles) with Eqs. (14) (green filled circles) and (15) (red filled). At $C=0.275$, in the top right we see that eigenvalues associated to the red $x$'s have moved inward towards zero. Next, for $C=0.3$, in the bottom left panel we observe that, after merging with zero, the eigenvalues now emerge from zero on the imaginary axis. Finally, at $C=0.7$ in the bottom right panel, we observe that, after the eigenvalues merge with the dashed-line intervals, an unstable quartet emerges from the axis.

Figure 16

The evolution of the soliton whose instability is predicted in the topmost left panel of Fig. 12 for $C=1$ and $\gamma =0.5$. The complex plane of all the eigenvalues for this solution is shown in the bottom left plot of Fig. 14. The top right plot shows the squared absolute values of perturbation amplitudes $a_n,c_n$ (higher amplitudes) and $b_n,d_n$ (lower amplitudes), defined in Eq. (4). The top left plot shows the solution at $t=0$, and the bottom left plot shows the solution at $t=122$ with $|{\Psi }_n{(t=122)|}^2$ in blue and $|{\Phi }_n{(t=122)|}^2$ in green. In the course of the evolution, the soliton maintains its shape, while the amplitude at the central rung $\left(n=1\right)$ grows with oscillations; the growth on the gain side, associated to ${\Psi }_n$, is ultimately dominant. Quantities $D_1\left(t\right)\equiv |{\Psi }_1{\left(t\right)|}^2-{\left|{\Psi }_1\left(0\right)\right|}^2$ and $D_2\left(t\right)\equiv |{\Phi }_1{\left(t\right)|}^2-{\left|{\Phi }_1\left(0\right)\right|}^2$ are shown in the bottom right plot, in order to better demonstrate the growing oscillations.

Figure 17

The evolution of the discrete soliton whose instability is predicted is in the topmost left panel of Fig. 12 for $C=1.5$ and $\gamma =0.5$. The complex plane of all the eigenvalues for this solution is shown in the bottom right plot of Fig. 14. The top right plot has the same meaning as in Fig. 16, where here $|b_n{|}^2,{\left|d_n\right|}^2$ are mostly zero while $|a_n{|}^2,{\left|c_n\right|}^2$ have nonzero amplitudes. In the course of the evolution, the soliton does not maintain its shape. In particular, the solution profiles at $t=22$ are shown in the left panel at the bottom bearing the apparent signature of the delocalized, unstable eigenmode; the delocalization is stronger in the $\Psi$ solution associated with gain. Similar to Fig. 16, we plot $D_1\left(t\right),D_2\left(t\right)$ in the bottom right plot. This plot shows that the central node experiences oscillations similar to Fig. 16, but the oscillatory effect is dominated by the delocalization seen in the bottom left plot, which grows and exceeds past the shorter peaks in the center.

Figure 18

The evolution of the solitary wave whose instability is predicted is in the middle left panel of Fig. 12 for $C=0.5$ and $\gamma =1.5$. The complex plane including all the eigenvalues for this solution is similar to the top right plot in Fig. 15. The top right plot here shows the squared absolute values of perturbation amplitudes $a_n,b_n$ (lower amplitudes) and $c_n,d_n$ (higher amplitudes) defined in Eq. (4). The top left plot shows the solution at $t=0$, and the bottom left plot shows the solution at $t=6.5$ with $|{\Psi }_n{\left(6.5\right)|}^2$ in blue and $|{\Phi }_n{\left(6.5\right)|}^2$ in green. In the course of the evolution, the soliton maintains its shape, while the amplitude at the rung $n=2$ grows with weak oscillations and with higher growth on the $\Psi$ side associated with gain. Quantities $D_1\left(t\right)\equiv |{\Psi }_1{\left(t\right)|}^2-{\left|{\Psi }_1\left(0\right)\right|}^2$ (blue line), $D_2\left(t\right)\equiv |{\Phi }_1{\left(t\right)|}^2-{\left|{\Phi }_1\left(0\right)\right|}^2$ (green line), $E_1\left(t\right)\equiv |{\Psi }_2{\left(t\right)|}^2-{\left|{\Psi }_2\left(0\right)\right|}^2$ (cyan line), and $E_2\left(t\right)\equiv |{\Phi }_2{\left(t\right)|}^2-{\left|{\Phi }_2\left(0\right)\right|}^2$ (red line) are shown in the bottom right plot to highlight the growth.