  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.
