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Similarity transformations for Nonlinear Schrodinger Equations with time varying coefficients: Exact results

In this paper we use a similarity transformation connecting some families of Nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. This transformation allows one to apply all known results for that equation to the original non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, using stationary solutions of the autonomous nonlinear Schrödinger equation we can construct exact breathing solutions to multidimensional non-autonomous nonlinear Schrödinger equations. An application is given in which we explicitly construct time dependent coefficients leading to solutions displaying weak collapse in three-dimensional scenarios. Our results can find physical applicability in mean field models of Bose-Einstein condensates and in the field of dispersion-managed optical systems.