## 3 Numerical Calculations and Analysis
### The Influence of Optical Lattice Parameters
However, when \(W_{0}=0.6\) (Figure 3c), the enhanced gain distribution results in a stronger lateral energy flow, leading to more pronounced oscillatory behavior. Figure 3d,e illustrate the variation in the peak intensity and centroid position of the Airyprime beam, when the propagation distance is at a fixed propagation distance of \(Z=30\), for different gain/loss distribution coefficients \(W_{0}\). It can be observed that, as \(W_{0}\) increases, the peak intensity of the beam gradually rises, while the centroid position undergoes periodic fluctuations. Notably, when \(W_{0}\) is positive (Figure 3c), the gain causes the beam's energy to propagate outward along the gain region. However, when \(W_{0}\) is negative (Figure 3a), the beam's energy tends to flow towards the loss region, resulting in backward propagation. Overall, the sign of \(W_{0}\) not only affects the beam's propagation direction but also determines its energy distribution and propagation stability. The larger the absolute value of \(W_{0}\), the more pronounced the propagation effects are, with increasing lateral shifts and fluctuations. The modulation frequency \(w\) is another factor that has a strong influence on the transmission characteristics of the soliton. At the modulation frequency \(w=0.8\) (Figure 4a), the shedding soliton has strong localization, the Airyprime beam is relatively concentrated and has small transverse oscillations which show relatively strong contraction during transmission, and the intensity of the shedding soliton is also larger. As the modulation frequency increases (Figure 4b,c), the peak power of the shed soliton decreases significantly. This is mainly because the attraction of the PT symmetry potential to the shedding soliton is weaker at this time, while the influence of the diffraction effect is relatively larger, resulting in the inability of the shedding soliton to maintain a stable shape. With increases in the modulation frequency \(w\), the attraction of the PT symmetric potential to the shedding soliton is gradually enhanced, which leads to the gradual concentration of the energy of the shedding soliton and a decrease in its oscillation amplitude. Therefore, the modulation frequency directly affects the distribution of the falling soliton in the spatial domain. A lower modulation frequency (e.g., \(w\) = 0.8) results in a stronger localization of the beam, leading to a smaller propagation range. In contrast, a higher modulation frequency (e.g., \(w\) = 1.2) causes the beam to spread more, weakening its localization and increasing its extent. Figure 4d,e illustrate the spatial variation in the peak intensity and the centroid position of the Airyprime beam under different modulation depths. It can be observed that, as the modulation frequency increases, the falling soliton exhibits a lower peak power and smaller lateral oscillations. These results indicate that the modulation frequency \(w\) has a significant impact on the transmission characteristics of the Airyprime beam. Figure 4: When \(a\) = 0.3, \(P\) = 1.9, and \(W_{0}\) = 0.8, the evolution of Airyprime beams with different modulation frequencies w is (**a**) \(w\) = 0.8, (**b**) \(w\) = 1, and (**c**) \(w\) =1.2. (**d**,**e**) are the corresponding peak power and center of gravity of the Airyprime beam when the normalized transmission distance \(Z\) = 30 and the modulation depth \(w\) = 0.8 (green line), \(w\) = 1 (orange line), and \(w\) = 1.2 (blue line), respectively. Figure 3: When \(a\) = 0.3, \(P\) = 2, and \(w\) = 1, the evolution of Airyprime beams with different gain/loss distribution coefficient \(W_{0}\) are (**a**) \(W_{0}\) = \(-\)0.6, (**b**) \(W_{0}\) = 0, and (**c**) \(W_{0}\) = 0.6. (**d**,**e**) are the corresponding peak power and center of gravity of the Airyprime beam when the normalized transmission distance \(Z\) = 30 and the modulation depth \(W_{0}\) = \(-\) 0.6 (green line), \(W_{0}\) = 0 (orange line), and \(W_{0}\) = 0.6 (blue line), respectively.