一类广义KdV行波系统的相图研究
Phase Diagrams of a Class of Generalized KdV Traveling Wave Systems
摘要: 广义KdV方程是由数百年前被提出的KdV方程延伸出来的一个方程，本文将研究其中一种形式。由于方程带有未知参数，我们将采取分类讨论的方法，通过变量替换和化简，把这一非线性偏微分方程转化为常微分方程组，从而得到该方程的行波系统。在对这一行波系统进行研究的过程中，我们需要讨论参数的取值对奇点分布位置的影响，并得到参数平面上的分支曲线。然后在分支曲线划分的不同区域下分别得到行波系统的分支相图。
Abstract: The generalized KdV equation is an extension of the KdV equation proposed hundreds of years ago. This paper will study one of its forms. Because the equation has unknown parameters, we will adopt the method of classified discussion through variable substitution and simplification, transform this non-linear partial differential equation into ordinary differential equations, and then obtain the traveling wave system of the equation. In the process of researching traveling wave system, we need to discuss the influence of parameter value on the location of singularity distribution, and get the bifurcation curve on the parameter plane. Then the branching phase diagrams of the traveling wave system are obtained under different regions of the branching curve.
文章引用：林桐. 一类广义KdV行波系统的相图研究[J]. 应用数学进展, 2019, 8(3): 503-511. https://doi.org/10.12677/AAM.2019.83056

参考文献

[1] Dey, B. (1986) Domain Wall Solutions of KdV-Like Equations with Higher Order Nonlinearity. Journal of Physics A, 19, L9-L12. [Google Scholar] [CrossRef] 
[2] Dey, B. (1988) KdV Like Equations with Domain Wall Solutions and Their Hamiltonians. In: Solitons, Springer, Berlin, 188-194.
[3] Zhang, J.F. (1998) New Solitary Wave Solution of the Combined KdV and mKdV Equation. International Journal of Theoretical Physics, 37, 1541-1546. [Google Scholar] [CrossRef] 
[4] Zhang, J.F., Wu, F.M. and Shi, J.Q. (2000) Simple Solition Solution Method for the Combined KdV and mKdV Equation. International Journal of Theoretical Physics, 39, 1697-1702. [Google Scholar] [CrossRef] 
[5] Yu, J. (2000) Exact Solitary Wave Solutions to a Combined KdV and mKdV Equation. Mathematical Methods in the Applied Sciences, 23, 1667-1670. [Google Scholar] [CrossRef] 
[6] Grimshaw, R., Peli-novsky, D., Pelinovsky, E. and Slunyaev, A. (2002) Generation of Large-Amplitude Solitons in the Extended Korteweg-de Vries Equation. Chaos, 12, 1070-1076. [Google Scholar] [CrossRef] [PubMed]
[7] Fan, E.G. (2002) Multiple Travelling Wave Solutions of Nonlinear Evolution Equations Using a Unified Algebraic Method. Journal of Physics A, 35, 6853-6872. [Google Scholar] [CrossRef] 
[8] Fan, E.G. (2003) Uniformly Con-structing a Series of Explicit Exact Solutions to Nonlinear Equations in Mathematical Physics. Chaos, Solitons and Fractals, 16, 819-839. [Google Scholar] [CrossRef] 
[9] Tang, M.Y., Wang, R.Q. and Jing, Z.J. (2002) Solitary Waves and Their Bifurcations of KdV Like Equation with Higher Order Nonlinearity. Science in China A, 45, 1255-1267.
[10] Peng, Y.-Z. (2003) New Exact Solutions to the Combined KdV and mKdV Equation. International Journal of Theoretical Physics, 42, 863-868. [Google Scholar] [CrossRef] 
[11] 刘正荣. 微分方程定性方法与数值模拟[M]. 广州: 华南理工大学, 2013.
[12] 宋明. 几类高次非线性波方程的行波解研究[D]: [博士学位论文]. 广州: 华南理工大学, 2014.

为你推荐