In the paraxial optical system, a beam propagating in a parabolic potential satisfies the normalized (1 + 1) D dimensionless linear Schrödinger equation [12]

$$i\frac{{\partial \psi }}{{\partial z}}+\frac{1}{2}\frac{{{\partial ^2}}}{{\partial {x^2}}}\psi \left( {x,z} \right) - \frac{1}{2}{\alpha ^2}{x^2}\psi \left( {x,z} \right)=0$$, (1)

where \(\psi \left( {x,z} \right)\) is the complex amplitude envelope of beam, x and z represent the transverse coordinate and propagation distance, respectively, \(\alpha\) is the depth of parabolic potential, which corresponds to the strong nonlocal nonlinearity. According to the Refs. [3, 12], the general solution of Eq. (1) with an arbitrary initial beam \(\psi \left( {x,0} \right)\) in real space can be written as \(\psi \left( {x,z} \right)=f\left( {x,z} \right)\int_{{ - \infty }}^{{+\infty }} {\left[ {\psi \left( {\xi ,0} \right){e^{ib{\xi ^2}}}} \right]} {e^{ - iK\xi }}d\xi\), (2)

where \(b=\frac{\alpha }{2}\cot \left( {\alpha z} \right)\), \(K=\alpha x\csc \left( {\alpha z} \right)\) and \(f\left( {x,z} \right)={e^{ib{x^2}}}\sqrt { - \frac{i}{{2\pi }}\frac{K}{x}}\).

One can see that the integral in Eq. (2) is a Fourier transform of \(\psi \left( {x,0} \right)\exp \left( {ib{x^2}} \right)\), with K being the spatial frequency. Then, according to the convolution property of Fourier transform, Eq. (2) can be rewritten as

$$\psi \left( {x,z} \right)=f\left( {x,z} \right)\frac{1}{{2\pi }}\int_{{ - \infty }}^{{+\infty }} {\widehat {\psi }} \left( {k,0} \right)\widehat {G}\left( {K - k} \right)dk$$, (3)

where \(\psi \left( {x,0} \right)\) and \(\widehat {G}\left( {k,0} \right)\) are the Fourier transform of \(\psi \left( {x,0} \right)\) and \({e^{ib{x^2}}}\), respectively. If we can calculate the Fourier transform of the initial beam, the analytical solution for arbitrary beams in a medium with parabolic potential can be easily obtained. To study the dynamic behavior of one-dimensional cosh-Gaussian beam, the input beam is taken as

$$\psi \left( {x,0} \right)={A_0}\cosh \left( {\Omega x} \right)\exp \left( { - \sigma {x^2}} \right)$$, (4)

where \({A_0}\) is a normalized coefficient to ensure that the input power is equal to 1. \(\cosh \left( \bullet \right)\) is the hyperbolic cosine function, \(\Omega\) is an initial parameter of the hyperbolic cosine function. \(\sigma\) is a parameter related to the width of Gaussian function.

Then, we can obtain the Fourier transforms of the input beam \(\psi \left( {x,0} \right)\) and \(G\left( x \right)=\exp \left( {ib{x^2}} \right)\) as

$$\widehat {\psi }\left( {k,0} \right)=\sqrt {\frac{\pi }{\sigma }} \exp \left( {\frac{{{\Omega ^2} - {k^2}}}{{4\sigma }}} \right)\cosh \left( {\frac{{i\Omega k}}{{2\sigma }}} \right)$$, (5)

$$\widehat {G}\left( {K,0} \right)=\sqrt {i\frac{\pi }{b}} \exp \left( { - \frac{i}{{4b}}{k^2}} \right)$$, (6)

Substituting Eqs. (5) and (6) into Eq. (3) for integral calculation, the analytical solution of Eq. (1) corresponding to the initial beam (4) can be derived as

$$\psi \left( {x,z} \right)=\sqrt {\frac{{i\pi }}{{b+i\sigma }}} f\left( {x,z} \right)\cosh \left[ {\frac{{\Omega K}}{{2\left( {b+i\sigma } \right)}}} \right]\exp \left[ {\frac{{i\left( {{\Omega ^2} - {K^2}} \right)}}{{4\left( {b+i\sigma } \right)}}} \right]$$, (7)

From Eq. (5), one can observe that the initial state is also a cosh-Gaussian wave packet in k space. In other words, the propagation of the beam in a parabolic potential is equivalent to a kind of automatic Fourier transform. In our numerical simulation, Eq. (1) is solved by means of a symmetrized split-step Fourier method, which assumes that the diffraction step and the nonlinear step separately. Generally speaking, the propagation properties of optical beam can be controlled by choosing appropriate parameters such as the depth of parabolic potential \(\alpha\), the linear chirp parameter and quadratic chirp parameter \(\beta\). Therefore, it is meaningful to discuss how these parameters affect the propagation properties of the cosh-Gaussian beam. Next, we will investigate the influence of the parabolic potential, the linear chirp parameter and quadratic chirp parameter on the dynamics of optical beams in details.