In this work, we present a new two-mode version to the complex Hirota equation. This new model arises in many applications in the field of optics, communications and other engineering sciences. It describes the propagation of two-symmetric solitary waves moving simultaneously with dependent phase-velocity parameter. The extended tanh-coth expansion method and the polynomial-function method are used to extract novel new cusp-type solutions to the Hirota model. A comprehensive graphical analysis is conducted to show physical aspects of this new type of nonlinear equations. The findings of this work are from mathematical point of view and we propose some benefits regarding the concept of two-mode where it could be visualized and practically applied.

中文翻译：

---

用于复杂 Hirota 模型的扩展双模版本的心尖和钟形尖峰光学孤子：在光学中的应用

在这项工作中，我们为复杂的 Hirota 方程提出了一个新的双模式版本。这种新模型出现在光学、通信和其他工程科学领域的许多应用中。它描述了与相速度参数相关的同时运动的双对称孤立波的传播。扩展的 tanh-coth 扩展方法和多项式函数方法用于为 Hirota 模型提取新的尖点类型解。进行了全面的图形分析以显示这种新型非线性方程的物理方面。这项工作的发现是从数学的角度来看的，我们提出了关于双模式概念的一些好处，它可以被可视化和实际应用。