In this paper, we introduce a new (4 + 1)-dimensional KdV-like equation. By using the Bell Polynomial method, we obtain the bilinear form, bilinear Bäcklund transformation, Lax pair and infinite conservation laws. It is proved that the equation is completely integrable in Lax pair sense. Based on the Hirota bilinear method and the test function method, high-order lump solutions, high-order lump-kink type N-soliton solutions, high-order lump-cosh-N-cos-M type soliton solutions, exp-cosh-N-cos-M type soliton solutions and exp-tanh-N-sin-M type soliton solutions (N,M →∞) for this equation are obtained with the help of symbolic computation. Via three-dimensional plots and contour plots with the help of Mathematics, analyses for the obtained solutions are presented, and their dynamic properties are discussed. Many dynamic models can be simulated by nonlinear evolution equations, and these graphical analyses are helpful to understand these models.

中文翻译：

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新的 (4 + 1) 维 KdV 类方程的可积性方面和一些丰富的解

在本文中，我们引入了一个新的 (4 + 1) 维 KdV-like 方程。通过使用贝尔多项式方法，我们得到了双线性形式、双线性 Bäcklund 变换、Lax 对和无限守恒定律。证明了该方程在Lax对意义上是完全可积的。基于广田双线性法和检验函数法，高阶块解，高阶块扭结型ñ-孤子解，高阶块-科什-ñ-因-米类型孤子解决方案，经验-科什-ñ-因-米类型孤子解决方案和经验-谭-ñ-罪-米类型孤子解决方案(ñ,米 →∞)因为这个方程是在符号计算的帮助下获得的。通过三维图和等高线图，借助数学，对得到的解进行分析，并讨论它们的动态特性。许多动态模型可以通过非线性演化方程来模拟，这些图形分析有助于理解这些模型。