# Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
## 1 Introduction  
An Airy beam exhibits a unique non-diffracting ability during the beam propagation [1] and can accelerate freely without any external help [2; 3]. In the event of obstacles, moreover, the Airy beam displays a self-healing property [4]. The evolution of the Poynting vector [5], the far-field divergent property [6], and the beam propagation factor [7], and the paraxiality [8] of an Airy beam have been investigated. Two methods of geometrical optics [9] and the Wigner distribution function [10] have been used to try to interpret the physical nature of the Airy beams. The propagation of Airy beams through an apertured misaligned paraxial \(ABCD\) optical system [11] and the fractional Fourier transform system [12] has been demonstrated previously. In addition, researches on the propagation of Airy beams in free space [13], water [14], turbulent atmosphere [15; 16; 17], turbulent ocean [18], uniaxial crystal [19], photonic lattice [20; 21], saturated medium [22], Kerr medium [23], nonlocal nonlinear media [24; 25], medium with parabolic potential [26], chiral materials [27], photorefractive media [28; 29], and inhomogeneous medium [30] have been reported. These investigations of Airy-beam propagation in different media have deepened the understanding of Airy beams and contributed to their effective utilization.  
The fascinating features of the Airy beam inspire researchers to propose new model beams originating from the Airy beams. In particular, circular Airy beams with autofocusing features [31; 32], generalized Airy beams [33; 34], and mirror and symmetrical Airy beams [35; 36] have been proposed. Here, we focus on two kinds of model beams, namely cosh-Airy and cos-Airy beams. A cosh-Airy beam can be obtained by the superposition of two Airy beams with different decay factors [37],while a cos-Airy beam is the outcome of the superposition of two linearly chirped Airy beams with positive and negative chirped coefficients. Based on the second-order moments, the beam propagation factor of a cosh-Airy beam has been investigated [38]. Although the cosh-Airy beams have similar propagation properties in free space as the Airy beams, they have more manipulation degrees of freedom than the corresponding Airy beams [37]. Also, the self-healing ability of the cosh-Airy beam is higher than that of the corresponding Airy beam [39]. Moreover, propagation properties of the cosh-Airy beam in uniaxial crystals orthogonal to the optical axis are verified to be more abundant than those of the corresponding Airy beam [40]. When a cosh-Airy beam propagates in a quadratic-index inhomogeneous medium, the periodic phase transition occurs [41]. However, to the best of our knowledge, there are very few research reports about cos-Airy beams. In the remainder of this paper, the propagation of cosh-Airy and cos-Airy beams in external parabolic potential is investigated. The cosh-Airy or cos-Airy beam will have two different optical field expressions in the general and the phase transition points. Moreover, the periodic behaviors of the cosh-Airy and cos-Airy beams in the parabolic potential are expected. This research helps to enrich the understanding of the beam characteristics of cosh-Airy and cos-Airy beams.