This paper extends the [Formula: see text]-dimensional Broer–Kaup equations in continuum mechanics to its fractional partner, which can model a lot of nonlinear waves in fractal porous media. Its derivation is demonstrated in detail by applying He’s fractional derivative. Using the semi-inverse method, two variational principles are established for the nonlinear coupled equations, which up to now are not discovered. The variational formulations can help to study the symmetries and find conserved quantities in the fractal space. The obtained variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be generalized to other nonlinear evolution equations with fractal derivatives.

中文翻译：

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具有分形导数的 (2 + 1)-维 BROER-KAUP 方程的变分原理

本文将连续介质力学中的[公式：见正文]维 Broer-Kaup 方程扩展到其分数伙伴，它可以模拟分形多孔介质中的大量非线性波。它的推导通过应用 He 的分数导数来详细说明。采用半反演方法，建立了非线性耦合方程的两个变分原理，这两个原理至今未被发现。变分公式有助于研究对称性并找到分形空间中的守恒量。通过变分法最小化泛函，证明了所获得的变分原理是正确的，并且可能在数值模拟中找到潜在的应用。该过程表明，半逆方法高效且强大，