Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto equation. Meanwhile, we find that this equation just exists in Lie point symmetries. Then, we present a conservation law, and derive reciprocal Bäcklund transformations of conservation law for the first time. Furthermore, Bäcklund transformation of KdV–Burgers–Kuramoto equation associated with truncated Painlevé expansion is studied for the first time. Subsequently, mCK method is employed to study KdV–Burgers–Kuramoto equation. Lastly, we investigate symmetries of generalized time fractional KdV–Burgers–Kuramoto equation, and derive a conservation law. The obtained results illustrate that symmetry method is a very powerful method to deal with nonlinear partial differential equations. The results provide theoretical support for explaining complex nonlinear phenomena.

中文翻译：

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广义 KdV-Burgers-Kuramoto 方程及其分数版本的对称性分析、解析解和守恒定律

波浪运动在流体动力学和工程问题中发挥着重要作用。本文采用对称方法对广义KdV-Burgers-Kuramoto方程进行了系统研究。首先，基于广义 KdV-Burgers-Kuramoto 方程的李点对称性，我们得到不变量和不变量解。特别是，我们获得了广义 KdV-Burgers-Kuramoto 方程的级数解。同时，我们发现这个方程只存在于李点对称中。然后，我们提出了一个守恒定律，并首次推导出守恒定律的倒数 Bäcklund 变换。此外，首次研究了与截断 Painlevé 展开相关的 KdV-Burgers-Kuramoto 方程的 Bäcklund 变换。随后，采用 mCK 方法研究 KdV-Burgers-Kuramoto 方程。最后，我们研究广义时间分数 KdV-Burgers-Kuramoto 方程的对称性，并推导出守恒定律。所得结果说明对称法是处理非线性偏微分方程的一种非常有效的方法。研究结果为解释复杂的非线性现象提供了理论支持。