## 3 Numerical Calculations and Analysis
### The Influence of Optical Lattice Parameters  
Figure 2 illustrates the spatial evolution of the Airyprime beam under different modulation depths in a PT symmetric optical lattice (Figure 2a-c), as well as the variations in the spatial and centroid positions concerning their propagation distance (Figure 2d,e). The modulation depth significantly influences the existence, stability, and transmission characteristics of the soliton. From the spatial evolution diagrams (Figure 2a-c), it can be seen that, due to the modulation of the PT potential, the main peak of the Airyprime beam is compressed in the initial stage. After propagating for a certain distance, a soliton with oscillatory transmission, where both its energy and width vary periodically, is shed. Figure 1: When \(P=1\) and \(W_{0}=1\), the contours of the real (green line) and imaginary (pink line) parts of the PT symmetric optical lattice for different \(w\) values are (**a**) \(w=1\), (**b**) \(w=-0.1\). Figure 2: When \(a=0.3\), \(w=1\), and \(W_{0}=1\), the evolution of Airyprime beams with different modulation depths is (**a**) \(P=1.5\), (**b**) \(P=2\), (**c**) \(P=2.5\). (**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 is \(P=1.5\) (green line), \(P=2\) (orange line), and \(P=2.5\) (blue line), respectively. A periodic oscillation occurs, causing the energy and width of the soliton to vary periodically. The rest of the soliton can recover its shape and self-acceleration along the parabolic trajectory due to its self-healing characteristics. Under the action of PT potential, the shedding soliton first moves to the right, and then reflects to the center after colliding with the boundary. This behavior indicates that the PT potential exerts an attractive force on the soliton, pulling it from the center towards the sides. However, upon reaching the boundary, an elastic collision occurs, altering the soliton's direction of motion. After this collision, the soliton moves toward the center, and upon reaching the center, it continues in its original direction. However, due to the modulation range of the PT potential primarily affecting the central region, the velocity of the soliton decreases as it moves further away from the center. Once the velocity reaches zero, the soliton reverses its direction and moves back toward the center. This repetitive motion mirrors the harmonic oscillation of a spring oscillator. Moreover, as the modulation depth increases, the transmission characteristics of the Airyprime beam undergo significant changes. As the modulation depth \(P\) increases (from 1.5 to 2.5), the beam's oscillation frequency accelerates, and the oscillation period decreases notably. Figures 2d and 2e, respectively, show the spatial variations in the peak intensity and the centroid position of the beam at different modulation depths. It can be observed that, with an increasing modulation depth, the soliton exhibits higher peak power and stronger transverse oscillations. These results indicate that the modulation depth \(P\) has a significant impact on the transmission characteristics of the Airyprime beam. Figure 3 illustrates the propagation evolution of Airyprime beams in a PT symmetric optical lattice under different gain/loss distribution coefficients, which are denoted as \(W_{0}\). Figure 3a-c show the spatial evolution of the Airyprime beam at different transmission distances when \(W_{0}\) is \(-0.6\), \(0\), and \(0.6\), respectively. It is evident that the evolution behavior of the Airyprime beam undergoes significant changes with variations in the gain/loss distribution coefficient \(W_{0}\). When \(W_{0}=-0.6\) (Figure 3a), the beam propagation is somewhat suppressed, and there is a reduced lateral energy flow. In contrast, when \(W_{0}=0\) (Figure 3b), the beam exhibits a more uniform energy distribution, and its propagation characteristics remain relatively stable.