应用(G'/G' + G + A)展开法求解mKdV方程的精确值解
Applying the (G'/G' + G + A) Expansion Method to Solve the Exact Value Solution of mKdV Equation
DOI: 10.12677/aam.2024.137316, PDF,   
作者: 翁琨锋, 邵廷朗：广东财经大学统计与数学学院，广东 广州
关键词: mKdV方程；G'/G' + G + A展开法；行波变换；mKdV Equation； G'/G' + G + A Expansion Method； Wave Transformation
摘要: 作为一个描述非线性波在具有极性对称性的系统中传播的模型，mKdV方程对于研究非线性光学中的波动问题等有重要的价值，对其作深入研究有利于物理光学中实际问题的解决，其求解方法的研究有着重要的意义。G'/G' + G + A展开法是近年来发展起来的基于齐次平衡原理的求解非线性偏微分方程的一种较为有效的方法。本文利用G'/G' + G + A展开法，运用行波变换，求解了mKdV方程，得到该方程的精确值解，并利用数学软件Maple画出了解的图像。
Abstract: As a model to describe the propagation of nonlinear waves in polar symmetric systems, mKdV equation is of great value to the study of wave problems in nonlinear optics. Further study of mKdV equation is beneficial to the solution of practical problems in physical optics, and the study of its solution method is of great significance. The G'/G' + G + A expansion method is an effective method for solving nonlinear partial differential equations based on the principle of homogeneous equilibrium developed in recent years. In this paper, mKdV equation is solved by using G'/G' + G + A expansion method and traveling wave transformation, and the exact value solution is obtained. Moreover, we use the mathematical software Maple to draw the understanding of the image.
文章引用：翁琨锋, 邵廷朗. 应用(G'/G' + G + A)展开法求解mKdV方程的精确值解[J]. 应用数学进展, 2024, 13(7): 3301-3308. https://doi.org/10.12677/aam.2024.137316

参考文献

[1] 陆求赐, 张宋传, 王学彬. 一类mKdV方程的孤立波解[J]. 中央民族大学学报(自然科学版), 2021, 30(2): 5-11.
[2] Ganie, A.H., Sadek, L.H., Tharwat, M.M., Iqbal, M.A., Miah, M.M., Rasid, M.M., et al. (2024) New Investigation of the Analytical Behaviors for Some Nonlinear PDEs in Mathematical Physics and Modern Engineering. Partial Differential Equations in Applied Mathematics, 9, Article ID: 100608. [Google Scholar] [CrossRef] 
[3] 李晓燕, 张英, 姚若侠. mKdV方程的一类新的精确孤立波解[J]. 宝鸡文理学院学报(自然科学版), 2013, 33(2): 1-4+8.
[4] 王思源, 陈浩. 求解KdV方程和mKdV方程的新方法: (g'/g2)展开法[J]. 华南师范大学学报(自然科学版), 2014, 46(1): 42-45.
[5] 李向正. 双函数展开法及mKdV方程的行波解[J]. 河南科技大学学报(自然科学版), 2013, 34(5): 82-86+9.
[6] 王鑫. 一类非线性偏微分方程的精确解[J]. 应用数学, 2013, 26(3): 521-525.
[7] Sahoo, S. and Saha Ray, S. (2016) Solitary Wave Solutions for Time Fractional Third Order Modified KdV Equation Using Two Reliable Techniques (G'/G)-Expansion Method and Improved (G'/G)-Expansion Method. Physica A: Statistical Mechanics and Its Applications, 448, 265-282. [Google Scholar] [CrossRef] 
[8] 陈继培, 陈浩. (g'/g2)展开法及其在耦合非线性Klein-Gordon方程中的应用[J]. 华南师范大学学报(自然科学版), 2012, 44(2): 63-66.
[9] Wen-An, L., Hao, C. and Guo-Cai, Z. (2009)The (ω/g)-Expansion Method and Its Application to Vakhnenko Equation. Chinese Physics B, 18, 400-404. [Google Scholar] [CrossRef] 
[10] Wang, M., Li, X. and Zhang, J. (2008) The (G'/G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423. [Google Scholar] [CrossRef] 
[11] Zhang, S., Dong, L., Ba, J. and Sun, Y. (2010) The (G'/G)-Expansion Method for a Discrete Nonlinear Schrödinger Equation. Pramana, 74, 207-218. [Google Scholar] [CrossRef] 
[12] Hong, B., Chen, W., Zhang, S. and Xu, J. (2019) The (G'/G' + G + A)-Expansion Method for Two Types of Nonlinear Schrödinger Equations. Journal of Mathematical Physics, 31, 1155-1156.
[13] 王明亮, 李志斌, 周宇斌. 齐次平衡原则及其应用[J]. 兰州大学学报, 1999(3): 8-16.

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