In recent years, how to realize effective modulation of optical beam and control the dynamics of beam in optics has become a key research topic for researchers. As a helpful method, the photonic potential is extensively investigated and employed in linear and nonlinear system. It is well known that different external potentials can be achieved by appropriately adjusting the refractive index of the material and various forms of potentials have been reported recently, such as linear potential [1], [2], parabolic potential [3], [4], [5], symmetric potential barrier [6], Gaussian potential [7], dynamic potentials [8], etc. However, among these potentials, the parabolic potential, extensively applied in laser-plasma physics [9], Bose–Einstein condensates [10], and ultracold atoms [11], is one of the most attractive and concerned potential mode. For example, the dynamics of finite-energy Airy beams characterized by non-diffraction and self-acceleration in a linear medium with a parabolic potential are discussed, whose results show that the beam undergoes periodic inversion, phase transition and unusual oscillation during propagation [12]. Meanwhile, the same team studied an anharmonic propagation of two-dimensional beams in a parabolic potential, in which periodic inversion and variable rotation are still there [13]. It should be noted that the parabolic potential has various impacts on the beam propagation, including Fourier transform operator [3], [14], [15], [16], modulation of a symmetric beam [17] and so on [18], [19]. Since self-Fourier transform can be associated with gradient-index (GRIN) media, the propagation of pulses in graded-index fibers can be achieved by introducing external parabolic potential in optics. In 2016, the propagation of Airy–Gaussian vortex beams through the gradient-index medium was studied with the transfer matrix method [20]. Besides, researchers investigated the propagation of the Pearcey Gaussian (PeG) beams in a medium with a parabolic refractive index, in which the parabolic refractive index is responsible for the focusing effect [21]. Subsequently, plenty of special beams in a medium with parabolic refractive index were studied, including cosh-Airy beam [17], Hermite–Gaussian vortex beam [22], chirped Pearcey Gaussian vortex beam [23] and so on [24]. Especially, strongly nonlocal nonlinear model proposed by Snyder and Mitchell [25] can be converted into the linear Schrödinger equation with a parabolic potential, thus we can conveniently simplify the nonlinear problem to a linear problem, which is why more and more related intriguing results were reported one after another [26], [27], [28], [29], [30], [31].

On the other hand, due to its auto-focusing properties, the cosh-Gaussian beams can be used in particle manipulation and many references have studied the propagation properties of such beams in different optical systems [32], [33]. Even though there are plentiful reports dealing with the parabolic potential, the propagation dynamics of the cosh-Gaussian beam with embedded first and second order chirped factors in a parabolic potential has not been discussed yet. In this paper, we investigate the propagation dynamics of on-axis and off-axis cosh-Gaussian beams in parabolic potential theoretically and numerically. The analytical expressions are derived by means of mathematical calculation and the analytical results are verified by numerical simulation.

The organization of this paper is as follows. In Section 2, we present a theoretical model of the propagation of cosh-Gaussian beams in parabolic potential and derive the corresponding analytical solution. In Section 3, we give the analytical and numerical results. In Section 4, we discuss the effect of initial chirp on the dynamics of the optical beam. Finally, the paper is concluded in Section 5.