The multiple exp-function method and multiple rogue wave solutions method are employed for searching the multiple soliton solutions to the (3 + 1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KP-Boussinesq-like) equation. The obtained solutions contain the first-order, second-order, third-order, and fourth-order wave solutions. At the critical point, the second-order derivative and Hessian matrix for only one point are investigated, and the lump solution has one maximum value. Moreover, we employ the linear superposition principle to determine N-soliton wave solutions of the KP-Boussinesq-like equation. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.

中文翻译：

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能量分布中出现的 (3 + 1) 维 Kadomtsev–Petviashvili–Boussinesq 类方程的多重流氓波解和线性叠加原理

多重exp函数法和多重流氓波解法被用于搜索(3+1)维Kadomtsev-Petviashvili-Boussinesq-like (KP-Boussinesq-like)方程的多重孤子解。得到的解包含一阶、二阶、三阶和四阶波解。在临界点，只研究了一个点的二阶导数和Hessian矩阵，块解有一个最大值。此外，我们采用线性叠加原理来确定类 KP-Boussinesq 方程的N 个孤子波解。通过选择合适的值，对这些获得的多孤子解的物理现象进行了分析和图示。