ABSTRACT Cnoidal wave and its extreme case, the solitary wave, can be described by the KdV equation, which was first derived by Korteweg and de Vires with the first-order accuracy. Subsequently, different authors proposed their derivations and claimed that their equations, sharing similar expressions but different corresponding coefficients, were the same as the original one. After introducing a unified dimensionless frame, this study re-derived the KdV equation with respect to seven existing methods and confirmed that KdV equation indeed refers to a type of first-order equations, rather than a specified one. Differences in equations come from the influence of the second-order quantities associated with the derivation process. Regarding the cnoidal wave and solitary wave, the KdV-type equations obtained using different methods present the same first-order solution for wave profile. Nevertheless, in their directly derived results, different wave celerities and water particle velocities are presented due to the influence of second-order quantities. Additionally, comparing with the second-order solutions, all directly derived wave celerity solutions predict well for the Ursell number between 20 and 100. As for the first-order solution of the water particle velocity, all methods present the same result except Dean’s expression which contains a different coefficient.

中文翻译：

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所有 KdV 类型的浅水波动方程都具有一致解吗？

摘要 Cnoidal 波及其极端情况，孤立波，可以用 KdV 方程描述，KdV 方程首先由 Korteweg 和 de Vires 以一阶精度导出。随后，不同的作者提出了他们的推导，并声称他们的方程，共享相似的表达式，但对应的系数不同，与原始方程相同。在引入统一的无量纲框架后，本研究针对现有的七种方法重新推导了 KdV 方程，并证实 KdV 方程确实指的是一类一阶方程，而不是特定的方程。方程的差异来自与推导过程相关的二阶量的影响。关于cnoidal波和孤立波，使用不同方法获得的 KdV 型方程给出了相同的波形一阶解。然而，在他们直接推导出的结果中，由于二阶量的影响，呈现了不同的波速和水粒子速度。此外，与二阶解相比，所有直接推导的波速解都对乌赛尔数在 20 到 100 之间进行了很好的预测。对于水粒子速度的一阶解，除了 Dean 表达式外，所有方法都得到了相同的结果包含不同的系数。