# Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
## 3 Numerical Simulations and Results  
First, the difference between cosh-Airy and cos-Airy beams is demonstrated in the input plane, which is shown in Figure 1. \(A=0.1\) in Figure 1. When \(\gamma=\beta\), the cosh-Airy beam has more side lobes than the cos-Airy beam. With increasing cosh factor, the number of the side lobes in the cosh-Airy beam also increases. With increasing cosine factor, however, the number of the side lobes in the cos-Airy beam decreases. Therefore, the cos-Airy beam is significantly different from the cosh-Airy beam. According to the formulae in the above section, the properties of cosh-Airy and cos-Airy beams in the parabolic potential are numerically simulated. In the following calculations, the depth of the potential and the decay factor remain unchanged, and they are set as \(L=0.5\) and \(a=0.1\), respectively. Figure 2 represents the contour graph of the normalized intensity of cosh-Airy beams with different transverse displacements in the parabolic potential. \(\gamma=0.04\) in Figure 2. The normalized intensity varies periodically upon propagation, and the period is \(T=4\pi\). The cosh-Airy beam with the positive transverse displacement accelerates opposite to it with the nonpositive transverse displacement. The curves of the normalized intensity distribution versus the propagation distance are diverse for different transverse displacements. The range of fluctuation of the main lobe in the case of the negative transverse displacement is far larger than that in the case of the contrary transverse displacement. When the transverse displacement disappears, the range of fluctuation of the main lobe is small during the forward-moving process. Figure 3 shows the contour graph of the normalized intensity of cosh-Airy beams with different cosh factors in the parabolic potential. \(x_{0}=10\) in Figure 3. When only the cosh factor increases, the number of side lobes adjacent to the main lobe increases significantly. The normalized intensity distribution of the cosh-Airy beam in different observation planes of the parabolic potential is shown in Figure 4, where \(\gamma=0.04\) and \(x_{0}=10\). When \(z=T/4\) and \(z=3T/4\), the normalized intensity takes on a symmetric Gaussian distribution, which is caused by the phase transition [26]. The phase transition points \(z=T/4\) and \(z=3T/4\) are also the inversion points of the intensity distribution. When \(0\leq z<T/4\), the side lobes are located at the left side of the main lobe. When \(T/4\leq z\leq 3T/4\), the side lobes are located at the right side of the main lobe, which is induced by the inversion of the intensity distribution. When \(3T/4<z\leq T\), the side lobes are again located at the left side of the main lobe, which is the result of two inversions of the intensity distribution. The blue curve in Figure 4 occurs after the red one. \(z=T/2\) is the inversion point of the accelerating direction. When \(0\leq z\leq T/2\), the cosh-Airy beam accelerates towards the negative direction of the \(x\)-axis. It should be emphasized that this direction of self-acceleration depends on the positive transverse displacement. When \(T/2\leq z\leq T\), the cosh-Airy beam accelerates towards the positive direction of the \(x\)-axis. Figure 1: Normalized intensity distribution of cosh-Airy (the red curve) and cos-Airy (the blue curve) beams with \(x_{0}=5\) in the input plane. Figure 3: Contour graph of the normalized intensity of cosh-Airy beams with different cosh factors in the parabolic potential. \(x_{0}=10\). (**a**) \(\gamma=0.02\). (**b**) \(\gamma=0.04\). (**c**) \(\gamma=0.06\). (**d**) \(\gamma=0.08\). Figure 2: Contour graph of the normalized intensity of cosh-Airy beams with different transverse displacements in the parabolic potential. \(\gamma=0.04\). (**a**) \(x_{0}=-5\). (**b**) \(x_{0}=0\). (**c**) \(x_{0}=5\). (**d**) \(x_{0}=10\). Figure 5 shows the contour graph of the normalized intensity of cos-Airy beams with different transverse displacements in the parabolic potential.