Exactly solvable stationary nonlinear Schrödinger equation duality family

Nonlinear Schrödinger equations can describe many-body interacting systems, such as Bose-Einstein condensates. We show the existence of a duality within stationary nonlinear Schrödinger equations, which is the same type as the Newton-Hooke duality in classical mechanics. This duality transforms one dynamical equation into another of the same type and simultaneously transforms the solution of the original equation into the solution of the transformed equation. That allows us to obtain from one solved dynamical equation the solution of another dynamical equation that is dual to it. Furthermore, we show the existence of the duality family, where any two members of a duality family are dual to each other. In a duality family, knowing the solution of one member allows the solutions of all other members to be obtained through duality transformations. The existence of the duality and duality family provides an approach to generating exactly solvable equations: starting with a solved equation, we can construct its duality family that contains an infinite number of exactly solvable equations by the duality transformation.