We consider the conservation laws, regarding the energy and mass, for the time-fractional nonlinear Schrödinger (TFNLS) equation on both continuous and discrete levels. By introducing a class of fractional Leibniz formulas on the complex-field, a local energy conservation law and a local mass conservation law are established for the TFNLS equation. The two conservative laws are asymptotically compatible with the associated conservation laws of the NLS equation in the fractional order limit \(\alpha \rightarrow 1^-\) . On the discrete level, by establishing the discrete counterparts of fractional Leibniz formulas, we build up a unified framework for the construction and analysis of conservative variable-step methods, which is suitable for many discrete Caputo formulas, including the variable-step L1, L1-2, L2, L2-1 \(_\sigma \) and L1 \(^+\) formulas. These variable-step methods are showed to admit a discrete local energy conservation and a discrete local mass conservation laws without restrictions. The discrete energy and the discrete mass conservation laws are all asymptotically compatible with the corresponding conservation laws of the associated variable-step scheme, respectively, for the NLS problem. Numerical experiments are provided to verify the accuracy and efficiency of the conservative methods together with an adaptive time-stepping strategy in long time simulations. So far as we know, this is the first work on the energy and the mass variation laws as well as their asymptotic compatibility for the TFNLS equation.
