Revertible and Symplectic Methods for the Ablowitz-Ladik Discrete Nonlinear Schroedinger Equation

摘要

Via coordinate transformations, the noncanonical symplectic structure of the Ablowitz-Ladik model (A-L model) of Nonlinear Schroedinger Equation (NLSE) can be standardized ([23]). We apply a symplectic and revertible scheme, a symplectic but irrevertible scheme and a non-symplectic but revertible scheme to the obtained standard Hamiltonian systems and directly to the nonstandardized A-L model, to simulate the solitons motion and test the evolution of the discrete invariants of the A-L model and also the conserved quantities of the original NLSE. In comparison with a higher-order non-symplectic and irrevertible scheme, we show the overwhelming superiorities of the symplectic methods and revertible methods. We also compare the implementation of the same symplectic or revertible scheme to different standardized Hamiltonian systems resulting from different coordinate transformations, and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one, in preserving the invariants of the A-L model and the original NLSE.