PT Meets Supersymmetry and Nonlinearity: An Analytically Tractable Case Example

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