Alaoui Mohammed Kbiri, Roshid Md Mamunur, Uddin Mahtab, Ma Wen-Xiu, Munshi Mohammod Jahirul Haque
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, 61413, Abha, Saudi Arabia.
Department of Mathematics, Hamdard University Bangladesh, Dhaka, Bangladesh.
Sci Rep. 2025 Apr 7;15(1):11923. doi: 10.1038/s41598-024-84941-9.
The perturbed Korteweg-de Vries (PKdV) equation is essential for describing ion-acoustic waves in plasma physics, accounting for higher-order effects such as electron temperature variations and magnetic field influences, which impact their propagation and stability. This work looks at the generalized PKdV (gPKdV) equation with an M-fractional operator. It uses bifurcation theory to look at critical points and phase portraits, showing system changes such as shifts in stability and the start of chaos. Figures 1, 2 and 3 provide detailed analyses of static soliton formation through saddle-node bifurcation. We also use the modified simple equation (MSE) method to look for ion-acoustic wave solutions directly, without having to first define them. This lets us find shapes like hyperbolic, exponential, and trigonometric waves. These solutions reveal complex phenomena, including double periodic waves, periodic lump waves, bright bell-shaped waves, and singular soliton waves. Additionally, we analyze modulation instability in the gPKdV equation, which signifies chaotic transitions and is crucial for understanding nonlinear wave dynamics. Those methods demonstrate their value in generating precise soliton solutions relevant to nonlinear science and mathematical physics. This research illustrates how theoretical mathematics and physics can support solutions to practical world issues, especially in energy and technological advancement. Fig. 1 The phase portraits of the system (5) for [Formula: see text]. Fig. 2 The two-dimensional phase portraits of the system (5) for [Formula: see text]. Fig. 3 The two-dimensional phase portraits of the system (9) for [Formula: see text].
扰动的科特韦格 - 德弗里斯（PKdV）方程对于描述等离子体物理学中的离子声波至关重要，它考虑了诸如电子温度变化和磁场影响等高阶效应，这些效应会影响离子声波的传播和稳定性。这项工作研究了带有M分数阶算子的广义PKdV（gPKdV）方程。它运用分岔理论来研究临界点和相图，展示了系统的变化，如稳定性的转变和混沌的起始。图1、图2和图3提供了通过鞍结分岔形成静态孤子的详细分析。我们还使用修正的简单方程（MSE）方法直接寻找离子声波解，而无需事先定义它们。这使我们能够找到诸如双曲线波、指数波和三角函数波等波形。这些解揭示了复杂的现象，包括双周期波、周期块状波、明亮的钟形波和奇异孤子波。此外，我们分析了gPKdV方程中的调制不稳定性，它标志着混沌转变，对于理解非线性波动力学至关重要。这些方法在生成与非线性科学和数学物理相关的精确孤子解方面显示出了价值。这项研究说明了理论数学和物理学如何能够支持解决实际世界中的问题，特别是在能源和技术进步方面。图1 系统（5）在[公式：见正文]时的相图。图2 系统（5）在[公式：见正文]时的二维相图。图3 系统（9）在[公式：见正文]时的二维相图。
