The fractional discrete nonlinear Schr\"odinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size $h>0$ and the nonlocal L\'evy index $\alpha \in (0,2]$. We show that the discrete system converges to the fractional NLS as $h \rightarrow 0$ below the energy space by directly estimating the difference between the discrete and continuum solutions in $L^2(\mathbb{T})$ using the discrete periodic Strichartz estimates. The sharp convergence rate via the finite difference method (FDM) is shown to be $O(h^{\frac{\alpha}{2+\alpha}})$ in the energy space. To further illustrate the convergent behavior of fDNLS, we survey various dynamical behaviors of the continuous wave (CW) solutions in the context of modulational instability, emphasizing the interplay between linear dispersion (or lattice diffraction), characterized by the nonlocal lattice coupling, and nonlinearity. In particular, the transition as $h \rightarrow 0$ from the linear dependence of maximum gain $\Omega_m$ on the amplitude $A$ of CW solutions to the quadratic dependence is shown analytically and numerically.

翻译：本文从解析与动力学角度，研究周期格点上分数阶离散非线性薛定谔方程（fDNLS）随网格尺寸$h>0$与非局部Lévy指标$\alpha \in (0,2]$变化的性质。通过运用离散周期Strichartz估计直接计算离散解与连续解在$L^2(\mathbb{T})$空间中的差异，我们证明当$h \rightarrow 0$时，离散系统在能量空间以下收敛于分数阶NLS。在能量空间中，有限差分法（FDM）给出的最优收敛速度为$O(h^{\frac{\alpha}{2+\alpha}})$。为深入阐明fDNLS的收敛行为，我们系统考察了连续波（CW）解在调制不稳定性背景下的多种动力学行为，重点分析了由非局部格点耦合表征的线性色散（或晶格衍射）与非线性效应之间的相互作用。特别地，我们通过解析与数值方法展示了当$h \rightarrow 0$时，最大增益$\Omega_m$对CW解振幅$A$的依赖关系如何从线性过渡到二次型。