Based on the investigation of (2+1)-dimensional ZK–mZK–BBM equation, it describes the gravity water waves in a long-wave regime. With the help of the semi-inverse method and the variational method, the time fractional ZK–mZK–BBM equation is derived in the sense of Riemann–Liouville fractional derivatives, which opens a new window for understanding the features of gravity water waves. Further, the symmetry of the (2+1)-dimensional time fractional ZK–mZK–BBM equation is studied by fractional order symmetry. Meanwhile, based on the new conservation theorem, the conserved laws of (2+1)-dimensional time fractional ZK–mZK–BBM equation are constructed. Finally, we show how to derive the solutions of the time fractional ZK–mZK–BBM equation by a bilinear method and the radial basis functions (RBFs) meshless approach.

中文翻译：

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重力波(2+1)维时间分数ZK-mZK-BBM方程的解、群分析和守恒定律

基于对(2+1)维ZK-mZK-BBM方程的研究，描述了长波状态下的重力水波。借助半反演法和变分法，推导出了黎曼-刘维尔分数阶导数意义上的时间分数ZK-mZK-BBM方程，为理解重力水波的特征打开了新的窗口。此外，通过分数阶对称性研究了(2+1)维时间分数ZK-mZK-BBM方程的对称性。同时，基于新的守恒定理，构造了(2+1)维时间分数ZK-mZK-BBM方程的守恒律。最后，我们展示了如何通过双线性方法和径向基函数 (RBF) 无网格方法推导时间分数 ZK–mZK–BBM 方程的解。