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In the present work, we combine the notion of $\mathcal{PT}$-symmetry withthat of super-symmetry (SUSY) for a prototypical case example with a complexpotential that is related by SUSY to the so-called P{\"o}schl-Teller potentialwhich is real. Not only are we able to identify and numerically confirm theeigenvalues of the relevant problem, but we also show that the correspondingnonlinear problem, in the presence of an arbitrary power law nonlinearity, hasan exact bright soliton solution that can be analytically identified and hasintriguing stability properties, such as an oscillatory instability, which thecorresponding solution of the regular nonlinear Schr{\"o}dinger equation witharbitrary power law nonlinearity does not possess. The spectral properties anddynamical implications of this instability are examined. We believe that thesefindings may pave the way towards initiating a fruitful interplay between thenotions of $\mathcal{PT}$-symmetry, super-symmetric partner potentials andnonlinear interactions.

PT Meets Supersymmetry and Nonlinearity: An Analytically Tractable Case Example