(1 + 1)-维Benjiamin Ono方程的行波解及分岔
Bifurcation and Travelling Wave Solutions for the (1 + 1)-Dimensional Benjiamin Ono Equation
DOI: 10.12677/AAM.2022.1111795, PDF,    科研立项经费支持
作者: 杨德牛：南昌应用技术师范学院，江西 南昌
关键词: 孤立波解；周期波解；非线性发展方程；分岔；Solitary Wave Solution； Periodic Wave Solution； Nonlinear Evolution Equation； Bifurcation
摘要: 本文借助平面动力系统分支理论和符号计算方法研究了(1 + 1)-维的Benjiamin Ono方程。首先经过行波变换得到二维的平面行波系统，借助Maple软件的符号计算得到了分岔的参数条件，同时给出了所有的分岔相图。利用平面行波系统的首次积分，把行波系统的求解转化为椭圆积分。然后讨论了在不同参数条件下所有精确解的解析表达式，包括周期波解、孤立波解，同时也给出了这些解的平面图像，通过图像可以很好的揭示其动力学行为。
Abstract: In the paper, the (1+1)-dimensional Benjiamin Ono equation is studied by means of bifurcation theory and method of plane dynamical systems. Firstly, the traveling wave system are obtained by traveling wave transformation. The parameter conditions of the bifurcation are obtained by the symbolic calculation of Maple software, and all the phase diagrams of the bifurcation are given. By using the first integral of a plane traveling wave system, the solution of the traveling wave system is transformed into an elliptic integral. Then, the analytical expressions of all the exact solutions un-der different parameters are discussed, including periodic wave solutions and solitary wave solu-tions. At the same time, the plane images of these solutions are given, which can reveal their dy-namic behaviors well.
文章引用：杨德牛. (1 + 1)-维Benjiamin Ono方程的行波解及分岔[J]. 应用数学进展, 2022, 11(11): 7503-7511. https://doi.org/10.12677/AAM.2022.1111795

参考文献

[1] McCue Scott, W., El Hachem, M. and Simpson, M.J. (2021) Exact Sharp-Fronted Travelling Wave Solutions of the Fisher-KPP Equation. Applied Mathematics Letters, 114, Article ID: 106918. [Google Scholar] [CrossRef] 
[2] 朱佐农. Fisher方程的新的显式精确孤波解[J]. 数学的实践与认识, 1995(2): 70-73.
[3] Yang, G.X. (2013) Hopf Bifurcation of Traveling Wave Solutions of Delayed Fisher-KPP Equation. Applied Mathematics and Computation, 220, 213-220. [Google Scholar] [CrossRef] 
[4] Matinfar, M. and Ghanbari, M. (2009) The Application of the Modified Variational Iteration Method on the Generalized Fisher’s Equation. Journal of Applied Mathematics and Com-puting, 31, 165-175. [Google Scholar] [CrossRef] 
[5] 张强, 曾艳, 周艳红. 一类广义Fisher方程的稳定性和动态分歧[J]. 四川大学学报(自然科学版), 2019, 56(2): 222-226.
[6] Kametaka, Y. (1976) On the Nonlinear Diffusion Equation of Kolmogorov-Petrovskii-Piskunov Type. Osaka Journal of Mathematics, 13, 11-66.
[7] Rodrigo, M. and Mimura, M. (2002) Annihilation Dynamics in the KPP-Fisher Equation. European Journal of Applied Mathematics, 13, 195-204. [Google Scholar] [CrossRef] 
[8] Hamel, F. and Roques, L. (2010) Fast Propagation for KPP Equations with Slowly Decaying Initial Conditions. Journal of Differential Equations, 249, 1726-1745. [Google Scholar] [CrossRef] 
[9] Moet, H.J.K. (2006) A Note on the Asymptotic Behavior of Solu-tions of the KPP Equation. SIAM Journal on Mathematical Analysis, 10, 728-732. [Google Scholar] [CrossRef] 
[10] Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York. [Google Scholar] [CrossRef] 
[11] 杨娟, 冯庆江, 曾春花. 一类非线性数学物理方程的精确解[J]. 数学的实践与认识, 2018, 48(17): 309-313.
[12] 许瑞麟, 许晓革, 孟祥花. (1+1)维Benjiamin Ono方程的精确显式解[J]. 北京信息科技大学学报(自然科学版), 2011, 26(3): 58-61.
[13] 陈立, 何姝琦, 董亚莹. Benny-Luke方程和Phi-4方程的精确行波解[J]. 纯粹数学与应用数学, 2015, 31(6): 604-610.
[14] 李继彬. 非线性非齐次弹性材料模型的精确模态波解和动力学性质[J]. 中国科学:数学, 2017, 47(1): 147-154.
[15] 李继彬. 两类Boussinesq方程的行波解分支[J]. 中国科学: 数学, 2008, 38(11): 1221-1234.

为你推荐