  In this paper, we propose a family of time-stepping schemes for approximating
general nonlinear Schr\"odinger equations. The proposed schemes all satisfy
both mass conservation and energy conservation. Truncation and dispersion error
analyses are provided for each proposed scheme. Efficient fixed-point iterative
solvers are also constructed to solve the resulting nonlinear discrete
problems. As a byproduct, an efficient one-step implementation of the BDF
schemes is obtained as well. Extensive numerical experiments are presented to
demonstrate the convergence and the capability of capturing the blow-up
phenomenon of the proposed schemes.
