Superpositions of bright and dark solitons supporting the creation of balanced gain-and-loss optical potentials

Bright and dark solitons of the cubic nonlinear Schrödinger equation are used to construct complex-valued potentials with all-real spectrum. The real part of these potentials is equal to the intensity of a bright soliton, whereas their imaginary part is defined by the product of such soliton with its concomitant, a dark soliton. Considering light propagation in Kerr media, the real part of the potential refers to the self-focusing of the signal and the imaginary one provides the system with balanced gain-and-loss rates.