## 3 Numerical Results
### Dipole Solitons
(**c**) is the phase structure of the dipole soliton. (**d**,**e**) correspond to the change of the gravity center, \(\Delta\), and the power of the soliton, \(P\), during propagation. (**f**) is the spectrum of the perturbation growth rate, where \(\mathrm{Im}(\lambda)\) and \(\mathrm{Re}(\lambda)\) are the imaginary part and the real part of the instability growth rate \(\lambda\). Figure 2: The power curves of dipole solitons in parity–time (PT)-symmetric triangular lattices with different nonlocality, \(d\), and degree of gain/loss, \(W_{0}\). The gray area in the \(\mu\)–\(W_{0}\) plane (propagation constant \(\mu\) versus degree of gain/loss \(W_{0}\)) is the optical energy band of the triangular lattices, thus the white area is the first bandgap. The solid lines indicate that the dipole solitons are stable, while the dashed lines indicate that they are unstable. means the right hump whose original position is (\(\pi\), 0) decays at the beginning of propagation and the power finally concentrates in the left hump. However, the power of this dipole soliton fluctuates violently during propagation, as shown in Figure 4e, which indicates this soliton is strongly unstable. The spectrum of \(\lambda\) is plotted in Figure 4f, demonstrating the linear instability of this dipole soliton. The maximum of the real part of \(\lambda\) is \(\max[\text{Re}(\lambda)]=0.0929\). When the nonlocality becomes stronger, dipole solitons with smaller \(\mu\) will be unstable, as shown by the red lines in Figure 1. We investigate the unstable dipole soliton with \(d=2\), \(\mu=7.6\) (the first row in Figure 5) and the stable one with \(d=2\), \(\mu=10\) (the second row in Figure 5). The evolution dynamics of these solitons are the same as that of the intermediate nonlocal type. For the unstable soliton, the right hump decays before \(z=40\), and its linear stability spectrum indicates it is linearly unstable. The corresponding maximum of \(\text{Re}(\lambda)\) is \(\max[\text{Re}(\lambda)]=0.0628\). For the stable soliton, the profile and power can be maintained before \(z=150\). Its gravity center suffers a slight oscillation, but the period of this oscillation is larger than the soliton with \(d=0.5\), \(\mu=9.6\). Further, the real part of \(\lambda\) is zero, which indicates this soliton is stable. In general, fewer multipole solitons can exist in the first gap with a larger degree of gain/loss, \(W_{0}\), as Figure 2 shows. To investigate the influence of \(W_{0}\) on the existence of multipole solitons, we calculate the maximum and minimum of propagation constants, \(\mu\), under which dipole solitons can exist with different \(W_{0}\). Three types of nonlocalities are considered separately, as shown in Figure 6. The green areas represent that dipole solitons can be found if their parameters are in these areas, and the shaded areas represent the optical energy band for linear optical modes where solitons cannot be found. With the weak nonlocality (\(d=0.05\)), dipole solitons exist in almost the whole first bandgap, as shown in Figure 6a. As the nonlocality becomes stronger, the existence range becomes narrower. With strong nonlocality (\(d=2\)), dipole solitons can only be found near the edge of the upper optical energy band, as shown in Figure 6c. Figure 4: The dynamics of the unstable dipole soliton with \(d=0.5,\mu=6\). (**a**–**c**) correspond to the transverse profile of solitons at \(z=0,z=40\), and \(z=150\), respectively. (**d**,**e**) correspond to \(\Delta\) and \(P\) during propagation, respectively. (**f**) shows the spectrum of the perturbation growth rate.