## 2 Theoretical Model  
The propagation of a beam in a PT-symmetric optical lattice is subject to medium loss and gain, which can be described by the normalized dimensionless nonlinear Schrodinger equation [22]:  
\[i\frac{\partial\varphi}{\partial z}+\frac{1}{2}\frac{\partial^{2}\varphi}{ \partial x^{2}}+P[V(x)+iW(x)]\varphi=0 \tag{1}\]  
In the formula, the parameter \(\varphi\) represents the field envelope distribution of the beam, the parameter \(z=Z/Z_{0}\) represents the normalized longitudinal coordinates along the propagation direction, the parameter \(x=X/X_{0}\) represents the normalized transverse coordinates, and the parameter \(P\) represents the modulation depth of the medium. In our study, \(X_{0}\) represents the initial width of the beam, which is typically related to the transverse characteristic scale of the beam. As for \(Z_{0}\), it is a normalized scaling length, defined as \(Z_{0}=\sqrt{Z_{d}\cdot Z_{nl}}\), where \(Z_{d}\) is the dispersion length and \(Z_{nl}\) is the nonlinear length. This length affects the strength of the complex potential. \(V(x)\) and \(W(x)\) are the real and imaginary parts of the dimensionless complex refractive index of the PT symmetric optical lattice, respectively. To satisfy the PT symmetry condition, the complex refractive index requires that the refractive index distribution of the optical lattice is an even function of the transverse spatial index, and that the gain/loss curve distribution is an odd function. That is, the real part and the imaginary part of the refractive index have parity symmetry. In this paper, the Gaussian PT symmetric structure optical lattice is used, the expression for which is as follows:  
\[V(x)=\exp\Bigl{(}-(\omega x)^{2}\Bigr{)} \tag{2}\]  
\[W(x)=W_{0}(\omega x)\exp\Bigl{(}-(\omega x)^{2}\Bigr{)} \tag{3}\]  
In the above formula, \(w\) is the modulation frequency of the complex potential, and \(W_{0}\) is the gain/loss distribution coefficient. The greater the modulation depth, the greater the intensity of the complex potential, and the stronger the modulation effect on the Airy beam. The modulation frequency determines the width and modulation range of the complex potential. The larger the modulation frequency, the narrower the complex potential and the smaller the modulation range of the beam. When the gain/loss distribution coefficient \(W_{0}>0\), the PT optical lattice generates gain in the \(x<0\) region, and the energy propagates to the gain region, resulting in the transverse flow of the beam to the right first. When \(W_{0}<0\), the gain is generated in the \(x<0\) regions, and the loss is generated in the \(x>0\) regions, resulting in the transverse flow of the beam to the left first. As the absolute value of \(W_{0}\) increases, the peak power and period of the soliton increase, which affects the propagation characteristics of the beam, including the generation, moving direction, transmission trajectory, and self-acceleration characteristics of the soliton. The initial input for the Airyprime beam can be written as follows:  
\[\varphi(z=0,x)=F(a)Ai^{\prime}(x)\exp(ax) \tag{4}\]  
In the formula, \(F(a)\) is introduced to keep the peak intensity of the beam at 1, 0 \(<a<\) 1 is the truncation coefficient, and \(Ai^{\prime}(x)\) is the first derivative of the Airy function. The expression of the Airy function is:  
\[Airy(x)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\exp[i(u^{3}/3+xu)]du \tag{5}\]  
The initial waveform of an Airyprime beam is an obvious asymmetric structure. The main peak is located in the second peak on the right side, and the first peak on the right side decays rapidly. There are multiple oscillation peaks on the left side of the main peak, and the oscillation amplitude gradually decreases, forming a long trailing structure.