The convergence of numerical method for nonlinear Schro¨dinger equation
Journal of Computational Mathematics
Symplectic methods for the nonlinear Schro¨dinger equation
Mathematics and Computers in Simulation - Special issue: solitons, nonlinear wave equations and computation
Numerical simulation of nonlinear Schro¨dinger systems: a new conservative scheme
Applied Mathematics and Computation
Symplectic methods for the Ablowitz-Ladik model
Applied Mathematics and Computation
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Via coordinate transformations, the noncanonical symplectic structure of the Ablowitz-Ladik model (A-L model) of Nonlinear Schrödinger 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.