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 periodic Strichartz estimates. The sharp convergence rate via the finite-difference method is shown to be $O(h^{\frac{\alpha}{2+\alpha}})$ in the energy space. On the other hand for a fixed $h > 0$, the linear stability analysis on a family of continuous wave (CW) solutions reveals a rich dynamical structure of CW waves due to the interplay between nonlinearity, nonlocal dispersion, and discreteness. The gain spectrum is derived to understand the role of $h$ and $\alpha$ in triggering higher mode excitations. The transition from the quadratic dependence of maximum gain on the amplitude of CW solutions to the linear dependence, due to the lattice structure, is shown analytically and numerically.

翻译：本文从解析与动力学角度研究了周期晶格上的分数阶离散非线性薛定谔方程(fDNLS)，通过改变网格尺寸$h>0$和非局部Lévy指数$\alpha \in (0,2]$。我们利用周期Strichartz估计，在$L^2(\mathbb{T})$中直接估计离散解与连续解的差异，证明了当$h \rightarrow 0$时，离散系统在能量空间以下收敛至分数阶NLS。通过有限差分方法，在能量空间中得到的尖锐收敛率为$O(h^{\frac{\alpha}{2+\alpha}})$。另一方面，对于固定的$h>0$，对一族连续波解进行的线性稳定性分析揭示了由于非线性、非局部色散和离散性之间的相互作用，连续波具有丰富的动力学结构。推导了增益谱以理解$h$和$\alpha$在激发更高模式中的作用。通过解析和数值方法证明了由于晶格结构，最大增益对连续波振幅的依赖关系从二次型转变为线性型。