The Schrödinger equation depends on the physical circumstance, which describes the state function of a quantum-mechanical system and gives a characterization of a system evolving with time. The essential focus of proposed research is to observe the solution for fractional generalized nonlinear Schrödinger (FGNS) equation using $q$-homotopy analysis transform method ( $q$-HATM). The fractional order derivative is taken in the Atangana-Baleanu (AB) sense. The physical behaviours of achieved solution for FGNS equation are discussed and sketch out graphically. The existence of the solution for the FGNS equation is presented through theorems 4.1 to 4.3. The proposed numerical simulations confirm the advantages of the AB derivative through $q$-HATM. Few numerical experiments were carried out to validate the proposed method. Moreover, numerical simulations are carried out to verify efficiency and robustness of the derived results by considering two cases.

中文翻译：

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利用非奇异导数传播经典光孤子的分数模型

薛定谔方程取决于物理环境，它描述了量子力学系统的状态函数，并给出了系统随时间演化的特征。拟议研究的重点是观察分数广义非线性薛定谔（FGNS）方程的解，使用$q$-同伦分析变换方法（ $q$-HATM）。分数阶导数采用 Atangana-Baleanu (AB) 意义。讨论了 FGNS 方程所获得解的物理行为，并以图形方式勾勒出来。FGNS 方程解的存在性通过定理 4.1 到 4.3 给出。所提出的数值模拟通过以下方式证实了 AB 导数的优势$q$-HATM。进行了一些数值实验来验证所提出的方法。此外，通过考虑两种情况，进行数值模拟以验证推导结果的效率和稳健性。