In this article, the exact wave structures are discussed to the Caudrey-Dodd-Gibbon equation with the assistance of Maple based on the Hirota bilinear form. It is investigated that the equation exhibits the trigonometric, hyperbolic, and exponential function solutions. We first construct a combination of the general exponential function, periodic function, and hyperbolic function in order to derive the general periodic-kink solution for this equation. Then, the more periodic wave solutions are presented with more arbitrary autocephalous parameters, in which the periodic-kink solution localized in all directions in space. Furthermore, the modulation instability is employed to discuss the stability of the available solutions, and the special theorem is also introduced. Moreover, the constraint conditions are also reported which validate the existence of solutions. Furthermore, 2-dimensional graphs are presented for the physical movement of the earned solutions under the appropriate selection of the parameters for stability analysis. The concluded results are helpful for the understanding of the investigation of nonlinear waves and are also vital for numerical and experimental verification in engineering sciences and nonlinear physics.

中文翻译：

---

CDG方程的多重exp函数，交叉扭结，周期扭结，孤立波方法和稳定性分析的应用

本文在基于Hirota双线性形式的Maple的帮助下，针对Caudrey-Dodd-Gibbon方程讨论了精确的波浪结构。研究表明该方程具有三角函数，双曲函数和指数函数解。我们首先构造一般指数函数，周期函数和双曲函数的组合，以导出该方程的一般周期扭解。然后，用更多的任意自变量参数给出了更多的周期波解，其中周期扭结解在空间的各个方向上都存在。此外，利用调制不稳定性讨论可用解的稳定性，并介绍了特殊定理。此外，还报告了约束条件，这些条件验证了解的存在性。此外，二维的曲线图是下稳定性分析的参数的适当选择提出了挣溶液的物理移动。结论有助于理解非线性波，对于工程科学和非线性物理学中的数值和实验验证也至关重要。