The present study is devoted to investigate the (2+1)-dimensional time-fractional long-wave dispersive (fLWD) model, which describes the dynamical behaviors of shallow water waves propagate along two horizontal directions with fast memories. We first establish the symmetries and conservation laws for this model by improving the fractional Lie group approach and fractional Noether’s formulae. Then, using the general fractional Erdélyi–Kober operator, we reduce the original equations into a time-fractional integro-differential system. Finally, the invariant subspace method is applied to this model to obtain more explicit solutions and the dynamical behaviors are analyzed through numerical simulations. This work extends the basic method to solve the higher dimensional system with mixed order of integer and fractional derivatives.

中文翻译：

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（2 + 1）维分数阶长波频散系统的不变分析，守恒律和一些精确解

本研究致力于研究（2 + 1）维时分长波色散（fLWD）模型，该模型描述了具有快速记忆功能的沿两个水平方向传播的浅水波的动力学行为。我们首先通过改进分数李群方法和分数Noether公式来建立该模型的对称性和守恒律。然后，使用一般的分数阶Erdélyi–Kober运算符，将原始方程式简化为时间分数阶积分微分系统。最后，将不变子空间方法应用于该模型以获得更明确的解，并通过数值模拟分析动力学行为。这项工作扩展了解决整数和分数阶导数混合阶数的高维系统的基本方法。