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On the \(L_\infty \) convergence of a difference scheme for coupled nonlinear Schrödinger equations. (English) Zbl 1198.65173

Summary: A finite difference scheme for coupled nonlinear Schrödinger equations is studied. The existence of the difference solution is proved by Brouwer fixed point theorem. With the aid of the fact that the difference solution satisfies two conservation laws, the finite difference solution is proved to be bounded in the discrete \(L_\infty \) norm. Then, the difference solution is shown to be unique and second order convergent in the discrete \(L\infty \) norm. Finally, a convergent iterative algorithm is presented.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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