This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted to a so-called gravitational optics. Besides Schrödinger–Newton equation, we have also concerned with linear and nonlinear Airy beam accelerations in flat and curved spaces and fractal photonics, related to nonlinear Schrödinger equation, where impact of the fractional Laplacian is discussed. Another important feature of the gravitational optics’ implementation is its geometry with the paraxial approximation, when quantum mechanics, in particular, fractional quantum mechanics, is an effective description of optical effects. In this case, fractional-time differentiation reflexes this geometry effect as well.

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引力光学中的分数阶薛定谔方程

本文解决了围绕分数量子力学概念的问题，这些问题与光在非均匀非线性介质中的传播有关，特别限于所谓的引力光学。除了薛定谔-牛顿方程外，我们还关注平面和弯曲空间中的线性和非线性艾里光束加速度以及与非线性薛定谔方程相关的分形光子学，其中讨论了分数拉普拉斯算子的影响。引力光学实现的另一个重要特征是它的几何具有近轴近似，当量子力学，特别是分数量子力学，是光学效应的有效描述时。在这种情况下，分数时间微分也反映了这种几何效应。