A nonlocal nonlinear Schr\"odinger model: well-posedness, local limit, and structure-preserving asymptotically compatible Fourier approximations
In this paper, we introduce a nonlocal nonlinear Schr\"odinger (NLS) model on periodic domains, and establish its well-posedness, conservation laws, local limit, dispersion properties, and develop a structure-preserving asymptotically compatible Fourier collocation method. We prove that for a fixed nonlocal horizon $\delta$, the model is globally well posed and conserves mass and nonlocal energy, and the nonlocal NLS solution converges to the local NLS solution with order $O(\delta^2)$ under suitable regularity assumptions. We also derive the dispersion relation and group velocity for plane waves, establish their $O(\delta^2)$ local limits, and characterize their behavior at high frequencies. The numerical method combines Crank--Nicolson time stepping with Fourier collocation and preserves mass and energy. The existence, uniqueness, and convergence of the numerical solutions are proved. In particular, the error of the nonlocal NLS solution is $O(\tau^2+N^{s-r})$ in the $H^s$ norm, uniformly with respect to the horizon. Moreover, its total $H^s$-error relative to the local NLS solution is $O(\delta^2+\tau^2+N^{s-r})$ without any coupling condition among $\delta$, $\tau$, and $N$, which proves asymptotic compatibility of the proposed method. Numerical experiments in one, two, and three dimensions are presented to verify the theoretical accuracy and discrete conservation, confirm convergence under independent variation of horizon and discretization parameters, and show how the horizon and kernel affect dispersive wave propagation.