Symplectic methods for the Ablowitz–Ladik discrete nonlinear Schrödinger equation

Abstract

Using several kinds of coordinate transformations, we standardize the noncanonical symplectic structure of the Ablowitz–Ladik model (A–L model) of nonlinear Schrodinger equation (NLSE), then we employ some symplectic scheme 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 method. We also compare the implementation of the same symplectic 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.