## 3 Numerical Calculations and Analysis
### The Influence of Airyprime Beam Parameters  
Figure 5 demonstrates the effect of the modulation of parameters of the PT symmetric optical lattice on the peak power and period of the shedding soliton of the Airyprime beam. Figure 5a shows that the peak power of the shedding soliton monotonically increases with increases in the modulation depth \(P\), while the period decreases with increases in \(P\). The peak power of the soliton oscillates faster than that of the modulation depth \(P\), and the period decreases with increases in \(P\). This implies that, the larger \(P\) is, the faster the soliton oscillates and the higher its energy. Figure 5b shows that the peak power and period of the shedding soliton show a U-shaped variation with the absolute value of the gain/loss distribution coefficient \(W_{0}\), and that the minimum value occurs at \(W_{0}=0\). This means that, the larger \(W_{0}\) is, the faster the soliton oscillates and the higher its energy. Figure 5c demonstrates that the peak power and period of the shedding soliton decrease monotonically with increases in the modulation frequency \(w\), indicating that, the higher \(w\) is, the weaker the soliton oscillation state is. By adjusting these parameters, precise control of the soliton properties can be realized, allowing adjustment of the speed, energy, and oscillatory state of the soliton. This is of great significance in studying the properties and applications of optical solitons. The truncation coefficient \(a\) is a degree of freedom that controls the waveform of the Airyprime beam. A numerical study was carried out to investigate the effect of the truncation coefficient on the propagation of the Airyprime beam in a PT symmetric optical lattice. The Airyprime beam exhibits an asymmetric tail-dragging oscillatory structure at the initial position, with the main peak having the highest energy. When propagating in a PT symmetric optical lattice, the main peak sheds an oscillatory soliton and forms a long trailing tail due to its self-recovery property (Figure 6a). The Airyprime beam exhibits an asymmetric multi-peak structure, with the truncation coefficient aaa controlling the level of side-lobe energy. As aaa increases, the side-lobe energy of the oscillatory tail decreases rapidly. Since the total energy remains constant, the energy from the tail side-lobes is transferred to the main peak (Figure 6b). When the truncation coefficient is increased to 0.4, the paraboloids of the beam gradually disappear, the beam almost evolves into a Gaussian shape, the main peak energy reaches a maximum, and the transverse self-accelerating property almost disappears (Figure 6c). Figure 6d,e show the variation in the peak intensity and the position of the center of gravity of the shedding soliton, respectively. The results show that, the larger the truncation coefficient \(a\) is, the higher the peak intensity of the shedding soliton is, and that the oscillation amplitude of the center of gravity position  
Figure 5: Effect of the parameters of the PT symmetric optical lattice—(a) modulation depth \(P\), (b) gain/loss distribution coefficient \(W_{0}\), and (c) modulation frequency \(w\)—on the peak power and period of the shedding soliton. Figure 3 illustrates the propagation evolution of Airyprime beams in a PT symmetric optical lattice. gradually increases with \(a\), indicating that the truncation coefficient \(a\) has a significant modulation effect on the energy distribution of the Airyprime beam as well as the peak intensity and center of gravity position of the shedding soliton. Figure 7 illustrates the effects of different truncation coefficients on the Airyprime beam. Figure 7a shows the spatial intensity distribution of the Airyprime beam at different truncation coefficients. It can be seen that the initial waveform of the Airyprime beam has an obvious asymmetric structure.