标题： 二维分数离散NLS方程：色散关系、孤立波、基本和涡旋孤子 显示英文标题

标题： Two-dimensional fractional discrete NLS equations: dispersion relations, rogue waves, fundamental and vortex solitons

摘要： 我们引入了二维（2D）分数晶格介质的物理相关的新型模型，考虑了分数位间耦合和位点自聚焦之间的相互作用。 我们的方法基于对连续Riesz分数导数的适当修改定义，提出了新颖的离散分数算子。 2D各向同性晶格模型使用了离散分数拉普拉斯算子，而2D各向异性系统则结合了沿正交方向独立作用的离散分数导数，这些方向具有不同的Lévy指数（LIs）。 我们推导出精确的线性色散关系（DRs），并识别出允许线性模式存在的频谱带，发现它们与连续对应项类似，除了波数范围有所不同。 此外，详细研究了离散模型中的调制不稳定性，并且与线性DRs类似，发现它与连续模型中的情况一致。 这种一致性突显了我们新定义的离散分数导数的性质。 此外，使用高斯输入，我们生成了各种异常波结构。 通过数值方法，我们系统地构建了2D基本和涡旋孤子族，并检查了它们的稳定性。 由于相互作用的离散性，基本孤子保持稳定，防止临界和超临界塌缩的发生。 另一方面，涡旋孤子在各向同性晶格模型中是不稳定的。 然而，在各向异性模型中——特别是其对称版本，其中两个方向上的LIs相等——保持了具有绕数$S=1$和$S=3$的稳定涡旋孤子。 详细结果强调了新定义的离散分数拉普拉斯算子在支持二维晶格介质中明确的孤子模式方面的鲁棒性。 显示英文摘要

摘要： We introduce physically relevant new models of two-dimensional (2D) fractional lattice media accounting for the interplay of fractional intersite coupling and onsite self-focusing. Our approach features novel discrete fractional operators based on an appropriately modified definition of the continuous Riesz fractional derivative. The model of the 2D isotropic lattice employs the discrete fractional Laplacian, whereas the 2D anisotropic system incorporates discrete fractional derivatives acting independently along orthogonal directions with different L\'evy indices (LIs). We derive exact linear dispersion relations (DRs), and identify spectral bands that permit linear modes to exist, finding them to be similar to their continuous counterparts, apart from differences in the wavenumber range. Additionally, the modulational instability in the discrete models is studied in detail, and, akin to the linear DRs, it is found to align with the situation in continuous models. This consistency highlights the nature of our newly defined discrete fractional derivatives. Furthermore, using Gaussian inputs, we produce a variety of rogue-wave structures. By means of numerical methods, we systematically construct families of 2D fundamental and vortex solitons, and examine their stability. Fundamental solitons maintain the stability due to the discrete nature of the interactions, preventing the onset of the critical and supercritical collapse. On the other hand, vortex solitons are unstable in the isotropic lattice model. However, the anisotropic one -- in particular, its symmetric version with equal LIs acting in both directions -- maintains stable vortex solitons with winding numbers $S=1$ and $S=3$. The detailed results stress the robustness of the newly defined discrete fractional Laplacian in supporting well-defined soliton modes in the 2D lattice media.