Fluids are seen in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid. On the basis of the Painlevé analysis, we find that the equation is Painlevé integrable under a certain constraint. Through the truncated Painlevé expansion, we give an auto-Bäcklund transformation. By virtue of the Hirota method, we derive a bilinear auto-Bäcklund transformation. Via the polynomial-expansion method, traveling-wave solutions are obtained. We observe that the amplitude of a traveling wave remains invariant during the propagation. We graphically demonstrate that the amplitude of the traveling-wave is affected by the coefficients corresponding to the dispersion and nonlinearity effects, while other coefficients have no influence on the traveling-wave amplitude, which represent the perturbed effects and disturbed wave velocity effects.

中文翻译：

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流体中 (3 + 1) 维广义 Kadomtsev-Petviashvili 方程的 Painlevé 分析、Bäcklund 变换和行波解

在大气科学、海洋学和天体物理学等学科中可以看到流体。在本文中，我们研究了流体中的 (3 + 1) 维广义 Kadomtsev-Petviashvili 方程。在Painlevé分析的基础上，我们发现方程在一定约束下是Painlevé可积的。通过截断的 Painlevé 展开，我们给出了一个自动 Bäcklund 变换。凭借 Hirota 方法，我们推导出了一个双线性自动 Bäcklund 变换。通过多项式展开法，得到行波解。我们观察到行波的幅度在传播过程中保持不变。我们以图形方式证明行波的幅度受与色散和非线性效应相对应的系数的影响，