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Nonautonomous integrable nonlinear Schrödinger equations with generalized external potentials

Inhomogeneous nonlinear Schrödinger equations are constructed systematically using the covariance with respect to the Darboux transformation. The Darboux invariants and the spectral parameter are functions of time and space, which result in equations describing nonautonomous solitons moving in generalized external potentials. All the known inhomogeneous nonlinear Schrödinger equations, as well as new ones of a more generalized form, can be derived from the constructed equation. One- and two-soliton solutions are explicitly constructed using the Darboux transformation.