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

GU Jieping; HUANG Wentao; CHEN Ting

doi:10.21656/1000-0887.410263

Volume 42 Issue 2

Feb. 2021

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2021 > 42(2): 221-232

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:

PDF( 448 KB)

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

doi: 10.21656/1000-0887.410263

¹College of Mathematics and Statistics, Guangxi Normal University,Guilin, Guangxi 541006, P.R.China;²School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, P.R.China

Funds: The National Natural Science Foundation of China（12061016；12001112）

Received Date: 2020-09-07
Rev Recd Date: 2020-09-23
Publish Date: 2021-02-01

Abstract

Abstract

The small-amplitude solitary periodic wave solutions and the local critical periodic bifurcations of the traveling wave equations for a class of reaction-diffusion equations with quintic nonlinear reaction terms and constant diffusion terms were studied. First, the reaction-diffusion equation was transformed into the corresponding traveling wave system through traveling wave transformation. The first 8 singular point quantities of the system were calculated with the singular point value method and the computer algebra software MATHEMATICA. Then, 2 center conditions for the singular point of the system were obtained, 8 limit cycles were proved to bifurcate at the origin of the traveling wave system, and 8 small-amplitude solitary periodic wave solutions were found to exist in the corresponding nonlinear reaction-diffusion equation. Furthermore, through computation of the period constants, the weak center order for the origin of the traveling wave system was derived. Then, the system was proved to have at most 3 local critical periodic bifurcations and be able to reach the 3 bifurcations. Moreover, the analysis of the critical periodic bifurcations of the traveling wave system reveals that the reaction-diffusion equation has 3 critical periodic wavelengths.
- reaction-diffusion equation,
- singular point quantity,
- limit cycle,
- solitary periodic wave,
- critical period bifurcation

FullText(HTML)

References(20)

References

[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.