In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

中文翻译：

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(3 + 1) 维非线性演化方程的可积性、呼吸器、流氓波、块、块多条带和块多孤子解

在本文中，我们考虑一个新的 (3 + 1) 维演化方程，它可用于解释非线性波在海洋中的传播。我们通过几个方面有效地研究了所考虑的非线性演化方程的可积性质。首先，我们给出了多维贝尔多项式理论的一些基本性质及其与 Hirota 双线性形式的关系。利用这些关系，我们推导出 Hirota 双线性形式和双线性 Bäcklund 变换。通过在双线性 Bäcklund 变换中使用 Cole–Hopf 变换，我们提出了一个 Lax 对。此外，利用贝尔多项式理论，我们计算了无数个守恒定律。此外，我们从 Hirota 双线性形式明确获得一、二和三孤子解，并以图形方式说明它们。通气解决方案还可以通过在二孤子解决方案中采用适当的复共轭参数来导出。选择由N孤子解导出的流氓波广义算法，我们直接获得一阶中心可控流氓波。通过采用完善的二次检验函数作为 Hirota 双线性形式的解来制定集总解。进一步采用二次函数和指数函数组合形式的测试函数，我们得到了块多条解。此外，二次和双曲余弦函数的组合形式产生了块多孤子解。以图形方式展示了块多条解和块孤子解演化中的裂变和聚变效应。