## 3 Numerical Results  
### Dipole Solitons  
We first consider dipole solitons, because they are the basic type of multipole solitons. We find that dipole solitons in PT-symmetric triangular lattices can be generated with different degrees of nonlocality, \(d\), and with different PT-symmetric potentials. As for PT-symmetric potentials, the primary parameter that we focus on is the degree of gain/loss, \(W_{0}\). The power curves of dipole solitons with different \(d\) and \(W_{0}\) are shown in Figure 2. It is intuitive that the nonlinear optical modes are bifurcated from the optical energy band. From the \(\mu\)-\(W_{0}\) plane, where \(\mu\) is the propagation constant, it is clear that the range of the first bandgap (i.e., the white area on the \(\mu\)-\(W_{0}\) plane) becomes narrower as \(W_{0}\) increases. Further, the first bandgap will disappear when \(W_{0}\) = 3, which means the PT symmetry is broken. With the same \(W_{0}\), dipole solitons can exist in weak nonlocality (\(d=0.05\)), intermediate nonlocality (\(d=0.5\)), and strong nonlocality (\(d=2\)). As the nonlocality becomes stronger, the power of dipole solitons is greater, which corresponds to a steeper slope as seen in Figure 2. However, fewer solitons can be found when \(W_{0}\) becomes greater. Generally speaking, stable dipole solitons tend to exist with weak nonlocality and small \(W_{0}\). In the following discussion, we choose the PT-symmetric triangular lattices with \(V_{0}\) = 2 and \(W_{0}\) = 0.3, of which the first bandgap is \(4.82<\mu<10.67\) and the infinite bandgap is \(\mu>10.75\). To investigate in detail the dynamics of the dipole solitons, we simulate their propagation. For the stable dipole solitons, they can maintain their profiles and power over a relatively long distance (\(z>100\)). We take the one with \(d=0.5,\mu=9.6\) as an example, as presented in Figure 3. This dipole soliton keeps its profile until \(z=140\), as shown in Figure 3a,b. Its phase structure is plotted in Figure 3c, which indicates that the two humps of this soliton are in-phase. We also calculate the change of the gravity center, which is defined as \(\Delta=\pm\sqrt{\left(x\prime-x_{0}\right)^{2}+\left(y\prime-y_{0}\right)^{2}}\), where \(\pm\) represents whether \(\left(x\prime,y\prime\right)\) is on the right (\(+\)) or left (\(-\)) of the initial position \(\left(x_{0},y_{0}\right)\). The gravity center suffers a slight oscillation before it is dispersed over the whole space, which is shown in Figure 3d. Physically, the oscillation of the gravity center is due to the redistribution of transverse power. Similar to the gravity center, the power of the dipole soliton also undergoes an oscillation. Once this soliton cannot maintain its profile, the nonlinearity of the medium will lead to a catastrophic focusing of the light, and then the power increases rapidly, which may destroy the medium, as shown in Figure 3e. The linear stability spectrum is shown in Figure 3f, and the real part of the growth rate \(\lambda\) is zero, which means that this dipole soliton is linearly stable against a small perturbation. Figure 1: The response of nonlinearity, \(R\), when (**a**) the degree of nonlocality (\(d\)) = 0.05, (**b**) \(d\) = 0.5, and (**c**) \(d\) = 5. In these figures, x refers to the horizontal coordinates. The insets of each picture are the corresponding transverse profiles of \(R\). The amplitude of each \(R\) is normalized. As for the unstable dipole solitons, we consider the dipole soliton with \(d=0.5,\mu=6\), as shown in Figure 4. The profiles of this soliton at \(z=0,40,150\) are plotted in Figure 4a-c, respectively. One hump of this dipole soliton decays quickly (before \(z=40\)), and another hump can maintain its profile over a long distance (z). From Figure 4d, it is obvious that the gravity center shifts from \((0,0)\) to \((-\pi,0)\), which  
Figure 3: The evolution dynamics of stable dipole soliton with \(d=0.5,\mu=9.6\). (**a**,**b**) correspond to the transverse profile of solitons at \(z=0\) and \(z=140\), respectively.