Nonlinear tunneling of solitons in a variable coefficients nonlinear Schr\"odinger equation with $\mathcal{P}\mathcal{T}$-symmetric Rosen-Morse potential

We construct soliton solution of a variable coefficients nonlinear Schr"odinger equation in the presence of parity reflection - time reversal $(\mathcal{P}\mathcal{T})-$ symmetric Rosen-Morse potential using similarity transformation technique. We transform the variable coefficients nonlinear Schr"odinger equation into the nonlinear Schr"odinger equation with $\mathcal{P}\mathcal{T}-$symmetric potential with certain integrability conditions. We investigate in-detail the features of the obtained soliton solutions with two different forms of dispersion parameters. Further, we analyze the nonlinear tunneling effect of soliton profiles by considering two different forms of nonlinear barrier/well and dispersion barrier/well. Our results show that the soliton can tunnel through nonlinear barrier/well and dispersion barrier/well with enlarged and suppressed amplitudes depending on the sign of the height. Our theoretical findings are experimentally realizable and might help to model the optical devices.