PT Meets Supersymmetry and Nonlinearity: An Analytically Tractable Case Example

Abstract

In the present work, we combine the notion of $\mathcal{P}\mathcal{T}$-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{ö}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{ö}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 $\mathcal{P}\mathcal{T}$-symmetry, super-symmetric partner potentials and nonlinear interactions.

References in corpus (4)

• Making Sense of Non-Hermitian Hamiltonians
• Visualization of Branch Points in PT-Symmetric Waveguides
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes

Cited by in corpus (14)

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• Novel PT-invariant Solutions For a Large Number of Real Nonlinear Equations
• Variational Approach to studying solitary waves in the nonlinear Schrodinger equation with Complex Potentials
• Phase-space quantum profile of Pöschl-Teller two-level systems
• Stability of new exact solutions of the nonlinear Schrodinger equation in a Poschl-Teller external potential
• Exact solutions of a generalized variant of the derivative nonlinear Schrodinger equation in a Scarff II external potential and their stability properties
• Symmetry breaking bifurcations and excitations of solitons in linearly coupled NLS equations with PT-symmetric potentials
• Stability of exact solutions of the nonlinear Schroedinger equation in an external potential having supersymmetry and parity-time symmetry
• Krein signature for instability of $\mathcal{P}\mathcal{T}$-symmetric states
• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry
• Soliton dynamics and stability in the ABS spinor model with a PT-symmetric periodic potential