Connected and disconnected stable regions of solitons of nonlinear Schrödinger equation with $\mathcal{PT}$-symmetric potential

Abstract: We have considered cubic nonlinear Schr\"odinger equation along with supersymmetric $\mathcal{PT}$ like potential and obtained exact stationary solutions in terms of bright and brigh-dark interacting solitons. The $\mathcal{PT}$ broken and $\mathcal{PT}$ unbroken regions are demonstrated also depicted. Connected and disconnected stable regions of bright and dark solitons are examined incorporating linear stability analysis validated by direct numerical simulations. Moreover, the strength of stability has been illustrated through excitations of bright and dark solitons.