Conservation laws and discrete counterparts for the time-fractional generalized nonlinear Schrödinger equation

Abstract

We study the conservation laws and structure-preserving numerical methods for the time-fractional generalized nonlinear Schrödinger (TFGNLS) equation. By introducing a fractional chain rule, we obtain a local energy conservation law which is asymptotically compatible with the energy conservation law of the generalized nonlinear Schrödinger (GNLS) equation. On the discrete level, by introducing a discrete fractional chain rule (DFCR), we build up a unified framework to establish the discrete local energy conservation law of variable-step methods without restrictions on time steps, which is suitable for any fractional backward differentiation formulas (FBDF) and fractional Crank–Nicolson (FCN) approximations of Caputo derivative. Numerical experiments are provided to verify the accuracy and efficiency of the variable-step L1 and L2 methods together with an adaptive time-stepping strategy in long time simulations.