Abstract In this article, we investigated a nonlinear higher dimensional time fractional Korteweg–de Vries-type (KdV) equation. This considered equation is usually used to describe shallow water waves phenomena in physics. Here we found firstly the symmetry of the higher dimensional time fractional KdV-type equation in the sense of the Riemann–Liouville (RL) fractional derivative with the aid of the fractional Lie symmetry method. Then, the one-parameter group of Lie point symmetry transformation and some special solutions of this considered equation, were obtained. Next, the optimal system of one-dimensional Lie subalgebra of this considered equation, was constructed. Subsequently, on the basis of the multiple-parameter Erdelyi–Kober fractional differential operator (FDO) and Erdelyi–Kober fractional integral operator (FIO), the original equation can be reduced into the lower dimensional fractional differential equation (FDE). Furthermore, it can be translated further into more lower dimensional FDE with new symmetry from the reduced equation. Finally, conservation laws of this discussed equation are also found through a new conservation theorem. These results are favorable support for us to understand this dynamic model in a deeper level.

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关于高维时间分数KdV型方程的可积性

摘要 在本文中，我们研究了非线性高维时间分数阶 Korteweg-de Vries 型 (KdV) 方程。这个考虑过的方程通常用于描述物理学中的浅水波现象。在这里，我们首先借助分数李对称方法发现了 Riemann-Liouville (RL) 分数阶导数意义上的高维时间分数阶 KdV 型方程的对称性。然后，得到了李点对称变换的单参数群和该方程的一些特解。接下来，构造了该方程的一维李子代数的最优系统。随后，在多参数 Erdelyi-Kober 分数阶微分算子（FDO）和 Erdelyi-Kober 分数阶积分算子（FIO）的基础上，原始方程可以简化为低维分数阶微分方程（FDE）。此外，它可以从简化的方程中进一步转换为具有新对称性的更低维的 FDE。最后，还通过新的守恒定理找到了该方程的守恒定律。这些结果为我们更深层次地理解这个动态模型提供了有利的支持。