A systematic construction of parity-time ($\cal PT$)-symmetric and non-$\cal PT$-symmetric complex potentials from the solutions of various real nonlinear evolution equations

TL;DR

This paper presents a systematic method to construct complex potentials, including PT-symmetric ones, for the Schrödinger equation using solutions from integrable nonlinear evolution equations, with applications to graphene models.

Contribution

It introduces a novel systematic approach to generate complex potentials from solutions of real nonlinear equations, expanding the class of exactly solvable models with potential applications.

Findings

Constructed various complex potentials including PT-symmetric types.

Linked the constructed potentials to graphene model applications.

Provided explicit forms of potentials from soliton and periodic solutions.

Abstract

We systematically construct a distinct class of complex potentials including parity-time ($\cal PT$) symmetric potentials for the stationary Schr\"odinger equation by using the soliton and periodic solutions of the four integrable real nonlinear evolution equations (NLEEs) namely the sine-Gordon (sG) equation, the modified Korteweg-de Vries (mKdV) equation, combined mKdV-sG equation and the Gardner equation. These potentials comprise of kink, breather, bion, elliptic bion, periodic and soliton potentials which are explicitly constructed from the various respective solutions of the NLEEs. We demonstrate the relevance between the identified complex potentials and the potential of the graphene model from an application point of view.