The well-known coupled Boussinesq–Burger equation can be used to describe the flow of the shallow water of the harbor. But when the boundary is nonsmooth, it becomes powerless. So, the fractal calculus is needed to be applied to it and the fractal coupled Boussinesq–Burger equation (FCBBe) is presented for the first time in this paper. By using the semiinverse method, we have successfully established the fractal variational formulation of the FCBBe, which can not only provide the conservation laws in an energy form in the fractal space but also reveal the possible solution structures of the equation. Then a novel variational approach based on He’s variational method and the two-scale transform are used to seek its periodic wave solutions. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. Finally, the numerical results have been shown through graphs to discuss the effect of different fractal orders on the wave motion. The obtained results in this work are expected to shed a bright light on the study of fractal nonlinear partial differential equations in the fractal space.

中文翻译：

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浅水分形耦合 BoussinesQ-Burger 方程研究的新视角

著名的耦合 Boussinesq-Burger 方程可用于描述港口浅水的流动。但是当边界不光滑时，它就变得无能为力了。因此，需要对其进行分形演算，本文首次提出了分形耦合Boussinesq-Burger方程（FCBBe）。利用半逆方法，我们成功地建立了FCBBe的分形变分形式，不仅可以提供分形空间中能量形式的守恒定律，还可以揭示方程可能的解结构。然后采用一种新的基于He变分法和两尺度变换的变分法求其周期波解。变分法的主要优点是可以降低微分方程的阶数，使方程更简单。最后，数值结果已通过图表显示，以讨论不同分形阶数对波浪运动的影响。本工作所取得的成果有望为分形空间中分形非线性偏微分方程的研究提供一盏明灯。