Analysis of a symplectic difference scheme for a coupled nonlinear Schrِdinger system

In general, proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations. In this paper, a symplectic difference scheme is proposed for an initial-boundary value problem of a coupled nonlinear Schrِdinger system. An important lemma and an induction argument are used to prove the unique solvability, convergence and stability of numerical solutions. An iterative algorithm is also proposed for the symplectic scheme and its convergence is proved. Numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis.