PT Meets Supersymmetry and Nonlinearity: An Analytically Tractable Case Example

TL;DR

This paper explores the integration of PT-symmetry and supersymmetry in complex potentials, revealing exact soliton solutions with unique stability features and potential implications for nonlinear quantum systems.

Contribution

It introduces a novel combination of PT-symmetry and supersymmetry in complex potentials, providing analytical solutions and stability analysis for nonlinear Schrödinger equations.

Findings

Eigenvalues confirmed numerically for the complex potential

Exact bright soliton solutions identified analytically

Discovery of oscillatory instability in soliton solutions

Abstract

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.