In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.

中文翻译：

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具有变系数和分数阶的扩展 Boussinesq-Whitham-Broer-Kaup 型方程的耦合分数行波解

在本文中，我们提出了具有可变系数和分数阶的扩展 Boussinesq-Whitham-Broer-Kaup (BWBK) 型方程。我们通过设置适当的平滑函数来考虑分数阶 BWBK 方程、分数阶 Whitham-Broer-Kaup (WBK) 方程和具有可变系数的分数阶 Boussinesq 方程，这些平滑函数是从所提出的方程导出的。我们通过采用改进的系统方法获得所考虑方程的均匀耦合分数行波解，随后它们的非对称行为以图形方式可视化。结果表明，改进的系统方法可以有效地求解分数阶非线性演化方程的显式行波解。