Investigating the dynamic characteristics of nonlinear models that appear in ocean science plays an important role in our lifetime. In this research, we study some features of the paired Boussinesq equation that appears for two-layered fluid flow in the shallow water waves. We extend the modified expansion function method (MEFM) to obtain abundant solutions, as well as to find new solutions. By using this newly modified method one can obtain novel and more analytic solutions comparing to MEFM. Also, numerical solutions via the Adomian decomposition scheme are discussed and favorable comparisons with analytical solutions have been done with an outstanding agreement. Besides, the instability modulation of the governing equations are explored through the linear stability analysis function. All new solutions satisfy the main coupled equation after they have been put into the governing equations.

中文翻译：

---

线性稳定性分析的新方法及其在海洋工程耦合Boussinesq方程中的应用

研究海洋科学中出现的非线性模型的动态特性在我们的一生中起着重要作用。在这项研究中，我们研究了成对的Boussinesq方程的一些特征，该方程对于浅水波中的两层流体流动而出现。我们扩展了改进的扩展函数方法（MEFM），以获得丰富的解决方案，以及寻找新的解决方案。通过使用这种新近修改的方法，与MEFM相比，可以获得新颖，更多的解析解。此外，讨论了通过Adomian分解方案的数值解，并与分析解进行了有利的比较，并取得了显着的共识。此外，通过线性稳定性分析函数来探索控制方程的不稳定性调制。