The core of this research is to investigate the system of equations for the ion sound and Langmuir waves through the Bernoulli sub-equation method. It is one of the most effective methods for solving nonlinear ordinary differential equations. To transform a given system to a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. Novel wave behaviors of the guiding system are acquired, such as kink-type singular solutions, wave solutions, periodic wave solutions, and periodic singular solutions. Besides, the modulation instability analysis of the given system is investigated based on the standard linear stability analysis and the modulation instability gain spectrum analysis. Furthermore, 2D, 3D and contour graphs of all acquired solutions are also plotted under the selection of appropriate parameter values.

中文翻译：

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离子声波和朗缪尔波方程组的调制不稳定性分析和解析解

本研究的核心是通过伯努利子方程法研究离子声和朗缪尔波的方程组。它是求解非线性常微分方程的最有效方法之一。为了将给定系统转换为非线性常微分方程，已经实现了行波转换。获得了导引系统的新型波动行为，例如扭结型奇异解，波动解，周期波动解和周期奇异解。此外，基于标准线性稳定性分析和调制不稳定性增益谱分析，研究了给定系统的调制不稳定性分析。此外，在选择合适的参数值的情况下，还绘制了所有获得的溶液的2D，3D和轮廓图。