AI 中文总结
本文针对周期域上的非局部非线性薛定谔模型，建立其适定性等理论结果，开发保结构渐近兼容傅里叶配置法，经多维数值实验验证了方法的精度与收敛性。
AI 中文摘要
本文在周期域上引入了非局部非线性薛定谔（NLS）模型，建立了其适定性、守恒律、局部极限、色散性质，并开发了一种保结构渐近兼容傅里叶配置法。我们证明，对于固定的非局部 horizon δ，该模型全局适定且守恒质量和非局部能量，在适当的正则性假设下，非局部 NLS 解以 O(δ²) 的阶收敛到局部 NLS 解。我们还推导了平面波的色散关系和群速度，建立了它们的 O(δ²) 局部极限，并表征了其高频行为。该数值方法结合了 Crank--Nicolson 时间步进与傅里叶配置，且守恒网格质量和原始网格能量。在时间步长 τ、傅里叶截断 N 以及精确解的 H^r 正则性假设下，非局部解的 H^s 误差为 O(τ² + N^{s-r})，且相对于 horizon 一致。此外，其相对于局部 NLS 解的总 H^s 误差为 O(δ² + τ² + N^{s-r})，且无需 δ、τ、N 之间的耦合条件，这证明了所提方法的渐近兼容性。本文给出了一维、二维和三维的数值实验，以验证理论精度和离散守恒性，确认 horizon 和离散化参数独立变化下的收敛性，并展示 horizon 和核如何影响色散波传播。
英文摘要
In this paper, we introduce a nonlocal nonlinear Schrödinger (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 $δ$, 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 $O(δ^2)$ under suitable regularity assumptions. We also derive the dispersion relation and group velocity for plane waves, establish their $O(δ^2)$ 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 $O(τ^2+N^{s-r})$ in the $H^s$ norm, uniformly with respect to the horizon. Moreover, its total $H^s$-error relative to the local NLS solution is $O(δ^2+τ^2+N^{s-r})$ without any coupling condition among $δ$, $τ$, 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.
Comments47 pages, 7 figures
