Abstract Solving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

中文翻译：

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一些著名方法与射影Riccati方程的关系

摘要 求解非线性演化方程是数学和物理科学中的一个重要问题。因此，求解非线性偏微分方程多采用特征法等传统方法。传统方法中还没有找到确定偏微分方程解析解的通用方法。由于符号计算技术的发展，在过去的 50 年中引入了许多替代方法，例如双曲正切函数方法。尽管它们都是作为一种新方法引入的，但其中一些方法彼此相似。在这项研究中，我们研究了以下四种在文献中大量使用的重要方法：tanh-coth 方法、改进的 Kudryashov 方法、F-展开法和广义Riccati方程映射法。这些方法的相似性引起了我们的注意，我们给出了这些方法与射影 Riccati 方程系统之间的联系。利用这种联系，可以推导出非线性演化方程的新求解方法。