# Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
## 3 Numerical Simulations and Results
\(\beta=0.04\) in Figure 5. When the sign of the transverse displacement changes, the direction of the self-acceleration for the cos-Airy beam also changes. Moreover, the range of fluctuation of the main lobe for the cos-Airy beam is more sensitive to the negative transverse displacement than the positive transverse displacement. A contour graph of the normalized intensity of cos-Airy beams with different cosine factors in the parabolic potential is shown in Figure 6, where \(x_{0}=10\). When only the cosine factor increases, the number of side lobes decreases. Figure 7 represents the normalized intensity distribution of the cos-Airy beam in different observation planes of the parabolic potential. \(\beta=0.04\) and \(x_{0}=10\) in Figure 7. Due to the positive transverse displacement, the cos-Airy beam in the first half-period accelerates towards the negative direction of the \(x\)-axis. In the latter half-period, however, the cos-Airy beam accelerates towards the positive direction of the \(x\)-axis. A Gaussian intensity profile emerges at the phase transition points \(z=T/4\) and \(z=3T/4\), which are also the inversion points of the intensity distribution. The main difference between Figures 4 and 7 is that the number of side lobes of the cosh-Airy beam is larger than that of the cos-Airy beam. In addition, the Gaussian intensity profile in Figure 3 is wider than that in Figure 7.  
Figure 8 shows the center of gravity of cosh-Airy beams in the parabolic potential. At first, the cosh factor is fixed, for example, \(\gamma=0.04\). The effect of the transverse displacement on the center of gravity is investigated here. When the transverse displacement is positive, the center of gravity in the first half-period decreases, while that in the latter half-period increases. When the transverse displacement is negative or zero, the center of gravity in the first half-period increases, while that in the latter half-period decreases. Figure 8 also denotes that the direction of the acceleration in the case of the positive transverse displacement is opposite to that in the case of the nonpositive transverse displacement. No matter what the transverse displacement is, \(z=T/2\) must be the inversion point of the accelerating direction. The amplitude of the center of gravity in the case of the negative transverse displacement is much larger than that in the case of the contrary transverse displacement. Then, the transverse displacement is fixed as \(x_{0}=5\), and the effect of the cosh factor on the center of gravity is investigated. The cosh factor only affects the amplitude of the center of gravity. As the cosh factor increases, the amplitude of the center of gravity decreases. The center of gravity of cos-Airy beams in  
Figure 1.0 Normalized intensity distribution of the cosh-Airy beam in different observation planes of the parabolic potential. \(\gamma=0.04\) and \(x_{0}=10\). the parabolic potential is shown in Figure 1.1 The effect of the transverse displacement on the center of gravity of the cos-Airy beam is similar to that of the cosh-Airy beam. The cosine factor only slightly affects the amplitude of the center of gravity. As the cosine factor increases, the amplitude of the center of gravity reduces slowly. The following conclusions can be drawn by comparing Figures 8a and 9a. When the transverse displacement is positive, the amplitude of the center of gravity of the cosh-Airy beam is smaller than that of the cos-Airy beam. When the transverse displacement is nonpositive, the amplitude of the center of gravity of the cosh-Airy beam is larger than that of the cos-Airy beam. Figure 1.2 Contour graph of the normalized intensity of cos-Airy beams with different transverse displacements in the parabolic potential. \(\beta=0.04\). (**a**) \(x_{0}=-5\). (**b**) \(x_{0}=0\). (**c**) \(x_{0}=5\). (**d**) \(x_{0}=10\). Figure 1.3 Contour graph of the normalized intensity of cos-Airy beams with different cosine factors in the parabolic potential. \(x_{0}=10\). (**a**) \(\beta=0.02\). (**b**) \(\beta=0.04\).