## 3 Numerical Results
### Six-Pole Solitons  
Another type of multipole solitons that we investigate is the six-pole solitons in the same PT-symmetric triangular lattices with nonlocal nonlinearity. Six-pole solitons with different collocations of \(d\) and \(W_{0}\) are found to exist, of which the power curves are plotted in Figure 7. The trends of these curves are as the same as the power curves of dipole solitons. However, we cannot find stable solitons with strong nonlocality (\(d=2\)), which is presented as the red dashed line in Figure 7. The power of six-pole solitons is about 3 times as large as the power of dipole solitons, because of the difference in the number of humps. For \(W_{0}=0.3\), the six-pole solitons with weak (\(d=0.05\)) or intermediate (\(d=0.5\)) nonlocality are stable enough to propagate without dispersion at \(z>140\). Moreover, all the power curves are bifurcated from the linear modes. Figure 5: The evolution dynamics of strongly nonlocal dipole solitons. The first row corresponds to the unstable soliton with \(d=2,\mu=7.6\), and the second row corresponds to the stable soliton with \(d=2,\mu=10\). (**a**,**d**) show their transverse profiles. (**b**,**e**) show the changes of the gravity center, \(\Delta\), during propagation. (**c**,**f**) show the spectrums of their perturbation growth rates. Figure 6: The existence range of dipole solitons in nonlocal nonlinear triangular lattices with different degrees of gain/loss. (**a**–**c**) correspond to weak (\(d=0.05\)), intermediate (\(d=0.5\)), and strong (\(d=2\)) nonlocalities, respectively. The shaded areas represent the optical energy band for linear optical modes. The green areas represent the parameter ranges in which dipole solitons can exist. The six-pole soliton with \(d=0.5,\mu=10\) is stable enough to propagate in the PT-symmetric triangular lattices. The profiles of this soliton at \(z=0\) and \(z=140\) are shown in Figure 8a,b, the structures of which are regular hexagons. The phase structure is plotted in Figure 8c, indicating its six humps are in-phase. We simulate the propagation of this soliton and calculate its gravity center and power in every step. The results are shown in Figure 8d,e. The transverse power distribution of this soliton is quite uniform, and thus the gravity center is almost at its original position. However, this six-pole soliton cannot maintain its profile after \(z=140\). This is because the nonlinearity will cause the oscillation of soliton power and increase the power catastrophically at a long distance. The result of linear stability analysis is plotted in Figure 8f. \(\mathrm{Re}(\lambda)\) is zero, demonstrating this six-pole soliton is linearly stable. In contrast to the intermediate nonlocality, the dynamics of six-pole solitons in the PT-symmetric triangular lattices with strong nonlocality are more complicated. These solitons may be concentrated into some specific points or create new hotspots. We present two unstable six-pole solitons with \(d=2,\mu=8.4\) (the first row in Figure 9) and \(d=2,\mu=10.4\) (the second row in Figure 9). For the six-pole soliton with \(d=2,\mu=8.4\), some humps of this soliton decay quickly, as shown in Figure 9a. Then, its power will shift into other humps, which increases the power density in the medium and may burn the medium. Figure 9b,c shows that the power of this soliton suffers a strong oscillation and eventually is concentrated into the left two points. The spectrum of the perturbation growth rate is plotted in Figure 9d, and \(\mathrm{max}[\mathrm{Re}(\lambda)]=0.0146\). For the six-pole soliton with \(d=2,\mu=10.4\), its initial profile is a hexagon with a weak hotspot in its center, as shown in Figure 9e. However, the power of that hotspot becomes stronger during propagation, which makes the soliton turn into seven humps, as shown in Figure 9f. From the evolution of power, which is plotted in Figure 9g, the power of this soliton is maintained until \(z=130\). Thus, the six humps of the soliton and the hotspot together average the initial power.