# Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
## 2 Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
 \tag{9}\]  
In the derivation of the above equation, the following integral formula is used [51]:  
\[\int_{-\infty}^{\infty}\exp\!\left(\frac{iu^{3}}{3}+ipu^{2}+iqu\right)\!du=2 \pi\exp\!\left[ip\!\left(\frac{2p^{2}}{3}-q\right)\right]\!Ai(q-p^{2}). \tag{10}\]  
When \(z=\left(m+\frac{1}{2}\right)\frac{\pi}{L}\), the one-dimensional cosh-Airy beam in the parabolic potential is described by:  
\[ \begin{array}{rl}\psi(x,z)=\frac{1}{2}\sqrt{-\frac{iK}{2\pi}}\exp(-iLx_{0}x) \left\{\exp\!\left(-a_{+}L^{2}x^{2}\right)\exp\!\left[\frac{a_{+}^{3}}{3}+ \frac{i\mathrm{s}}{3}(L^{3}x^{3}-3a_{+}^{2}Lx)\right]\right.\\ &\qquad\qquad\qquad\left.+\exp\left(-a_{-}L^{2}x^{2}\right)\exp\!\left[\frac{ a_{+}^{2}}{3}+\frac{i\mathrm{s}}{3}(L^{3}x^{3}-3a_{-}^{2}Lx)\right]\right\},\end{array}  \tag{11}\]  
where \(s=1\) if \(m\) is even, and \(s=-1\) if \(m\) is odd. When \(z=\left(m+\frac{1}{2}\right)\frac{\pi}{L}\), the intensity distribution of one-dimensional cosh-Airy beam in the parabolic potential turns out to be:  
\[ \begin{array}{rl}\left|\psi(x,z)\right|^{2}=\frac{L}{8\pi}\!\left\{\exp(-2a_ {+}L^{2}x^{2})\exp\!\left(\frac{2a_{+}^{3}}{3}\right)+\exp(-2a_{-}L^{2}x^{2}) \exp\!\left(\frac{2a_{-}^{3}}{3}\right)\right.\\ &\qquad\qquad\left.+2\exp[-(a_{+}+a_{-})L^{2}x^{2}]\exp\!\left(\frac{a_{+}^{2} }{3}+\frac{a_{-}^{3}}{3}\right)\cos[(a_{+}^{2}-a_{-}^{2})Lx]\right\}.\end{array}  \tag{12}\]  
When \(z=m\pi/L\), \(\sin(Lz)=0\). In this case, \(b=\infty\) and \(K=\infty\). When \(z\)\(m\pi/L\), Equation (9) reduces to:  
\[\psi(x,z)=\frac{-1}{\sqrt{s}}Ai(sx-x_{0})\exp[a(sx-x_{0})\cosh[\gamma(sx-x_{0}]. \tag{13}\]  
Finally, the input beam is a cos-Airy beam, which is characterized by:  
\[\psi(x,0)=Ai(x-x_{0})\exp[a(x-x_{0})]\cos[\beta(x-x_{0})], \tag{14}\]where \(\beta\) is the cosine factor. A cos-Airy beam is the result of the superposition of two linearly chirped Airy beams with chirped coefficients \(\beta\) and \(-\beta\). When \(z\
eq\left(m+\frac{1}{2}\right)\frac{\pi}{L}\), the general solution of one-dimensional cos-Airy beam in the parabolic potential is found to be:  
\[ \begin{array}{ll}\psi(x,z)=&-\frac{1}{2}\sqrt{\frac{K}{2\hbar x}}\exp(ibx^{2 })\left\{\exp\left[\frac{\left(a+i\beta\right)^{3}}{3}\right]Ai\left(\frac{K} {2b}-\frac{1}{16b^{2}}-\frac{\beta}{2b}-x_{0}+\frac{ia}{2b}\right)\right.\\ &\times\exp\left[\left(a+i\beta+\frac{i}{4b}\right)\left(\frac{K}{2b}-\frac{1} {16b^{2}}-\frac{\beta}{2b}-x_{0}+\frac{ia}{2b}\right)\right]\exp\left[-\frac{ iK^{2}}{4b}-\frac{1}{3}\left(a+i\beta+\frac{i}{4b}\right)^{3}\right]\\ &+\exp\left[\frac{\left(a-i\beta\right)^{3}}{3}\right]Ai\left(\frac{K}{2b}- \frac{1}{16b^{2}}+\frac{\beta}{2b}-x_{0}+\frac{ia}{2b}\right)\exp\left[-\frac {iK^{2}}{4b}-\frac{1}{3}\left(a-i\beta+\frac{i}{4b}\right)^{3}\right]\\ &\times\exp\left[\left(a-i\beta+\frac{i}{4b}\right)\left(\frac{K}{2b}-\frac{1} {16b^{2}}+\frac{\beta}{2b}-x_{0}+\frac{ia}{2b}\right)\right].\end{array}  \tag{15}\]  
When \(z=\left(m+\frac{1}{2}\right)\frac{\pi}{L}\), the one-dimensional cos-Airy beam in the parabolic potential turns out to be:  
\[ \begin{array}{ll}\psi(x,z)=&\frac{1}{2}\sqrt{-\frac{iK}{2\pi}}\exp(-iLx_{0} x)\exp(-aL^{2}x^{2})\left\{\exp\left[\frac{\left(a+i\beta\right)^{3}}{3}-i \beta L^{2}x^{2}+\frac{i\beta}{3}\left(L^{3}x^{3}-3a^{2}Lx\right)\right]\\ &\qquad\qquad\qquad+\exp\left[\frac{\left(a-i\beta\right)^{3}}{3}+i\beta L^{2 }x^{2}+\frac{i\beta}{3}\left(L^{3}x^{3}-3a^{2}Lx\right)\right]\right\}.\end{array}