This study investigates the Gilson–Pickering equation by using the sine-Gordon expansion method. Sine-Gordon expansion method is one of the most powerful methods for solving the nonlinear partial differential equations. We successfully construct various exact solitary wave solutions to the governing equation, such as shock wave, topological, non-topological, compound topological, and non-topological soliton wave solutions. In addition, the stability of the studied nonlinear equation is analyzed via the linear stability analysis. The 2D, 3D, and contour surfaces are also plotted for all obtained solutions.

中文翻译：

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等离子体物理学中的吉尔森-皮克森方程的新波行为和稳定性分析

这项研究使用正弦-戈登展开法研究了吉尔森-皮克森方程。Sine-Gordon展开法是求解非线性偏微分方程的最有效方法之一。我们成功地为控制方程构造了各种精确的孤立波解，例如冲击波，拓扑，非拓扑，复合拓扑和非拓扑孤子波解。另外，通过线性稳定性分析来分析所研究的非线性方程的稳定性。还为所有获得的解决方案绘制了2D，3D和轮廓表面。