Stability and response of trapped solitary wave solutions of coupled nonlinear Schr\"odinger equations in an external, $\mathcal{P}\mathcal{T}$- and supersymmetric potential

We present trapped solitary wave solutions of a coupled nonlinear Schr"odinger system in $1$+$1$ dimensions in the presence of an external, supersymmetric and complex $\mathcal{P}\mathcal{T}$-symmetric potential. The Schr"odinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated by means of analytical and numerical methods. Two different variational approximations are considered where the stability and dynamics of the solitary waves are explored in terms of eight and twelve time-dependent collective coordinates. We find regions of stability for specific potential choices as well as analytic expressions for the small oscillation frequencies in the collective coordinate approximation. Our findings are further supported by performing systematic numerical simulations of the nonlinear Schr"odinger system.