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Numerical simulation of a nonlinearly coupled Schrödinger system: A linearly uncoupled finite difference scheme. (English) Zbl 1202.65116
A system of strong coupled 1D nonlinear Schrödinger equations in an arbitrary large limited in space and time domain at appropriate initial-boundary value conditions is considered. For solving the problem a conservative finite difference scheme is constructed, based on a regular three-level nine-point stencil. The scheme is uncoupled in computation for the two unknown discrete functions. It is shown that the difference scheme conserves the discrete energy and discrete masses in conformity with the corresponding properties of the differential problem. Using the energy method unconditional stability and second-order convergence of the scheme in the $$L_2$$-norm are proved.
Numerical tests are given to demonstrate advantages of the proposed scheme in comparison with the two-level conservative scheme obtained by W. J. Sonnier and C. I. Christov [Math. Comput. Simul. 69, No. 5–6, 514–525 (2005; Zbl 1119.65396)] and the multi-symplectic scheme by J.-Q. Sun and M.-Z. Qin [Comput. Phys. Commun. 155, No. 3, 221–235 (2003; Zbl 1196.65195)].

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations
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