留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

一类反应扩散方程的孤立周期波和局部临界周期分支

古结平 黄文韬 陈挺

古结平, 黄文韬, 陈挺. 一类反应扩散方程的孤立周期波和局部临界周期分支[J]. 应用数学和力学, 2021, 42(2): 221-232. doi: 10.21656/1000-0887.410263
引用本文: 古结平, 黄文韬, 陈挺. 一类反应扩散方程的孤立周期波和局部临界周期分支[J]. 应用数学和力学, 2021, 42(2): 221-232. doi: 10.21656/1000-0887.410263
GU Jieping, HUANG Wentao, CHEN Ting. Solitary Periodic Waves and Local Bifurcations of Critical Periods for a Class of Reaction-Diffusion Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 221-232. doi: 10.21656/1000-0887.410263
Citation: GU Jieping, HUANG Wentao, CHEN Ting. Solitary Periodic Waves and Local Bifurcations of Critical Periods for a Class of Reaction-Diffusion Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 221-232. doi: 10.21656/1000-0887.410263

一类反应扩散方程的孤立周期波和局部临界周期分支

doi: 10.21656/1000-0887.410263
基金项目: 国家自然科学基金(12061016;12001112);广西自然科学基金(重点项目)(2016GXNSFDA380031);广西研究生教育创新计划项目(YCSW2020105)
详细信息
    作者简介:

    古结平(1996—),男,硕士生(E-mail: gujieping3032@163.com);黄文韬(1966—),男,教授,博士生导师(通讯作者. E-mail: huangwentao@163.com);陈挺(1989—),男,博士(E-mail: chenting0715@126.com).

  • 中图分类号: O175.12

Solitary Periodic Waves and Local Bifurcations of Critical Periods for a Class of Reaction-Diffusion Equations

Funds: The National Natural Science Foundation of China(12061016;12001112)
  • 摘要: 研究了一类含有五次非线性反应项和常数扩散项的反应扩散方程的小振幅孤立周期波解,以及它的行波方程局部临界周期分支问题.运用行波变换将反应扩散方程转换为对应的行波系统,应用奇点量方法和计算机代数软件MATHEMATICA计算出该系统的前8个奇点量,得到该系统奇点的两个中心条件,并证明行波系统原点处可分支出8个极限环,对应的非线性反应扩散方程存在8个小振幅孤立周期波解;通过周期常数的计算,得到了行波系统原点的细中心阶数,并证明该系统最多有3个局部临界周期分支,且能达到3个局部临界周期分支;通过分析行波系统的临界周期分支,得到该反应扩散方程有3个临界周期波长.
  • [1] ARANSON I, KRAMER L. The world of the complex Ginzburg-Landau equation[J]. Review of Modern Physics,2001,74(1): 99-143.
    [2] CHEN A Y, GUO L N, DENG X J. Existence of solitary waves and periodic waves for a perturbed generalized BBM equation[J]. Journal of Differential Equations,2016,261(10): 5324-5349.
    [3] ZHUANG K G, DU Z J, LIN X J. Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method[J]. Nonlinear Dynamics,2015,80(1/2): 629-635.
    [4] MANSOUR M B A. Traveling wave solutions of a reaction-diffusion model for bacterial growth[J]. Physica A: Statistical Mechanics and Its Applications,2007,383(2): 466-472.
    [5] SHERRATT J A, SMITH M J. Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models[J]. Journal of the Royal Society Interface,2008,5(22): 483-505.
    [6] LI J B, WU J H, ZHU H P. Traveling waves for an integrable higher order KdV type wave equations[J]. International Journal of Bifurcation and Chaos,2006,16(8): 2235-2260.
    [7] MANOSA V. Periodic travelling waves in nonlinear reaction-diffusion equations via multiple Hopf bifurcation[J]. Chaos, Solitons and Fractals,2003,18(2): 241-257.
    [8] HUANG W T, CHEN T, LI J B. Isolated periodic wave trains and local critical wave lengths for a nonlinear reaction-diffusion equation[J]. Communications in Nonlinear Science and Numerical Simulation,2019,74(5): 84-96.
    [9] SANCHEAGARDUNO F, MAINI P K. Traveling wave phenomena in some degenerate reaction-diffusion equations[J]. Journal of Differential Equations,1995,117(2): 281-319.
    [10] YANG G X. Hopf bifurcation of traveling wave solutions of delayed Fisher-KPP equation[J]. Applied Mathematics and Computation,2013,220(4): 213-220.
    [11] CHICONE C, JACOBS M. Bifurcation of critical periods for plane vector fields[J]. Transactions of the American Mathematical Society,1989,312(2): 433-486.
    [12] ROMANOVSKI V G, HAN M A. Critical period bifurcations of a cubic system[J]. Journal of Physics A: Mathematical and General,2003,36(18): 5011-5022.
    [13] ROUSSEAU C, TONI B. Local bifurcations of critical periods in the reduced Kukles system[J]. Canadian Journal of Mathematics,1997,49(2): 338-358.
    [14] YU P, HAN M A. Critical periods of planar revertible vector field with third-degree polynomial functions[J]. International Journal of Bifurcation and Chaos,2009,19(1): 419-433.
    [15] LIU Y R, LI J B. Theory of values of singular point in complex autonomous differential systems[J]. Science in China (Series A),1990,33: 10-24.
    [16] 黄文韬. 微分自治系统的几类极限环分支与等时中心问题[D]. 博士学位论文. 长沙: 中南大学, 2004. (HUANG Wentao. Several classes of bifurcations of limit cycles and isochronous centers for differential autonomous systems[D]. PhD Thesis. Changsha: Central South University, 2004. (in Chinese))
    [17] LIU Y R, HUANG W T. A new method to determine isochronous center conditions for polynomial differential systems[J]. Bulletin des Sciences Mathématiques,2003,127(2): 133-148.
    [18] YU P, HAN M A. Twelve limit cycles in a cubic case of the 16th Hilbert problem[J]. International Journal of Bifurcation and Chaos,2005,15(7): 2191-2205.
    [19] CHEN H B, LIU Y R. Linear recursion formulas of quantities of singular point and applications[J]. Applied Mathematics and Computation,2004,148(1): 163-171.
    [20] GEYER A, VILLADELPRAT J. On the wave length of smooth periodic traveling waves of the Camassa-Holm equation[J]. Journal of Differential Equations,2015,259(6): 2317-2332.
  • 加载中
计量
  • 文章访问数:  1634
  • HTML全文浏览量:  456
  • PDF下载量:  314
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-09-07
  • 修回日期:  2020-09-23
  • 刊出日期:  2021-02-01

目录

    /

    返回文章
    返回