Symplectic Numerical Methods For The Ablowitz-Ladik Discrete Nonlinear Schrodinger Equation:Case Of Dark Solitons Motion

In this paper,firstly,we discrete the original nonlinear Schrodinger equation(NLSE).Then we use several kinds of coordinate transformations,including symmetric coordinate transformation and non-symmetric one,to standardize the non-canonical symplectic structure of the Ablowitz-Ladik model(A-L model)of NLSE.Secondly,we employ some symplectic,symmetric,non-symplectic and non-symmetric schemes 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 scheme applied directly to the A-L model,we show the overwhelming superiorities of the symplectic ones.We also compare the implementation of different schemes including symplectic,symmetric,non-symplectic and non-symmetric schemes to the same standardized Hamiltonian system and show that the symmetric schemes or the symplectic schemes improves the numerical results obtained via the non-symmetric and non-symplectic one,in preserving the invariants of the A-L model and the original NLSE.