Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

In this paper we obtain a similarity transformation connecting nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. As applications we construct exact breathing solutions to multidimensional non-autonomous nonlinear Schrödinger equations and discuss how to construct time-dependent coefficients leading to solutions which collapse weakly in three-dimensional scenarios. Our results are applicable to the study of the dynamics of Bose–Einstein condensates in the mean-field limit and dispersion-managed optical systems.