Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time and reverse space-time nonlocal integrable equations, including nonlocal NLS, nonlocal sine-Gordon equation and nonlocal Davey-Stewartson equation etc. And, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS which is direct discretization version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for NLS equation.problem of this equation is solved under special initial values by inverse scattering transform (IST). But for general initial values, the solutions of Cauchy problem of nonlocal NLS equation can not be obtained by using IST. In our study, we investigate a nonintegrable space discrete nonlocal NLS equation. We obtain its stationary solitary wave solutions by discrete Fourier transform. The linear stability of these stationary solitary wave solutions are examined. Then, numerical simulations for Cauchy problem of nonlocal NLS equation with periodic boundary conditions are performed, in which we use the integrable scheme and the nonintegrable scheme as a spatial discrete model of nonlocal NLS equation. We find some different and new properties on the numerical solutions for Cauchy problem of nonlocal NLS equation comparing with the numerical solutions of the Cauchy problem for the NLS equation.

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We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term.