JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (8): 88-94.doi: 10.6040/j.issn.1671-9352.0.2021.615

Previous Articles     Next Articles

Bifurcation phenomena and nonlinear wave solutions of BKK equation

HAN Qing-xiu, LIU Hong-xia, WU Yun*   

  1. School of Mathematics Science, Guizhou Normal University, Guiyang 550025, Guizhou, China
  • Online:2022-08-20 Published:2022-06-29

PDF (PC)

331

Abstract: The bifurcation phenomena and nonlinear wave solutions of Broer-Kaup-Kupershmidt(BKK)equation are studied. Firstly, the first integral and singularities of the BKK equation are obtained by travelling wave transformation. Then the bifurcation phase portraits of the BKK equation in each region are presented by using the bifurcation method and qualitative theory of dynamical systems, and some nonlinear wave solutions of the equation are obtained. Furthermore, three bifurcation phenomena of kink wave are revealed. Finally, Maple software is used to simulate these bifurcation phenomena.

Key words: Broer-Kaup-Kupershmidt equation, nonlinear wave solution, bifurcation phenomenon

CLC Number: 

  • O175.29
[1] ARSHAD M, SEADAWY A R, LU D, et al. Travelling wave solutions of Drinfeld-Sokolov-Wilson, Whitham-Broer-Kaup and(2+1)-dimensional Broer-Kaup-Kupershmit equations and their applications[J]. Chinese Journal of Physics, 2017, 55(3):780-797.
[2] XU Guiqiong, LI Zhibin. Exact travelling wave solutions of the Whitham-Broer-Kaup and Broer-Kaup-Kupershmidt equations[J]. Chaos, Solitons and Fractals, 2004, 24(2):549-556.
[3] 翁献君,杨先林.(2+1)维Broer-Kaup-Kupershmidt方程的精确解[J]. 通信技术,2008,41(10):216-217. WENG Xianjun, YANG Xianlin. The exact solution of(2+1)-dimensional Broer-Kaup-Kupershmidt equation[J]. Communication Technology, 2008, 41(10):216-217.
[4] 陈未路,张雯婷,张李溥,等.(2+1)维Broer-Kaup-Kupershmidt方程dromions之间的相互作用(英文)[J]. 理论物理通讯,2013,59(1):5. CHEN Weilu, ZHANG Wenting, ZHANG Lipu, et al. Interaction behaviors between special dromions in the(2+1)-dimensional Broer-Kaup-Kupershmidt equation[J]. Theoretical Physical Communication, 2013, 59(1):5.
[5] YOUNIS M, RIZVI S, BALEANU D, et al. Lump and rogue wave solutions for the Broer-Kaup-Kupershmidt system[J]. Chinese Journal of Physics, 2020, 68:19-27.
[6] WEN Xiaoyong. N-soliton solutions and localized structures for the(2+1)-dimensional Broer-Kaup-Kupershmidt system[J]. Nonlinear Analysis Real World Applications, 2011, 12(6):3346-3355.
[7] KASSEM M M, RASHED A S. N-solitons and cuspon waves solutions of(2+1)-dimensional Broer-Kaup-Kupershmidt equations via hidden symmetries of lie optimal system-science-direct[J]. Chinese Journal of Physics, 2019, 57:90-104.
[8] TANG Y, YUEN M, ZHANG L. Double wronskian solutions to the(2+1)-dimensional Broer-Kaup-Kupershmidt equation[J]. Applied Mathematics Letters, 2020, 105:106285.
[9] YOMBA E. The extended Fans sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations[J]. Physics Letters A, 2005, 336(6):463-476.
[10] 王晓民,苏道毕力格,庞晶.(2+1)维Broer-Kaup-Kupershmidt方程组的行波解[J]. 数学的实践与认识,2015, 45(16):203-208. WANG Xiaoming, SUDAO Bilige, PANG Jing. Traveling wave solution of(2+1)-dimensional Broer-Kaup-Kupershmidt equations[J]. Practice and Knowledge of Mathematics, 2015, 45(16):203-208.
[11] 张芸芝,江波.(2+1)维Broer-Kaup-Kupershmidt方程的同宿轨道与孤立波解[J]. 江苏技术师范学院学报,2012,18(4):64-67. ZHANG Yunzhi, JIANG Bo. Homoclinic orbit with isolated wave solutions of(2+1)-dimensional Broer-Kaup-Kupershmidt equation[J]. Journal of Jiangsu Technical Normal College, 2012, 18(4):64-67.
[12] WU Yun, LIU Zhengrong. New types of nonlinear waves and bifurcation phenomena in Schamel-Korteweg-de Vries equation[J/OL]. Abstract and Applied Analysis, 2013[2022-1-10].https://doi.org/10.1155/2013/483492.
[13] WU Yun, LIU Zhengrong. Bifurcation phenomena of nonlinear waves in a generalized Zakharov-Kuznetsov equation[J/OL]. Advances in Mathematical Physics, 2013[2022-1-10].https://doi.org/10.1155/2013/812120.
[14] SONG Ming, LI Shaoyong, CAO Jun. New exact solutions for the(2+1)-dimensional Broer-Kaup-Kupershmidt equations[J/OL]. Abstract and Applied Analysis, 2010[2022-1-10].https://doi.org/10.1155/2010/652649.
[15] 唐亚宁,陈燕南.(2+1)维Broer-Kaup-Kupershmidt方程的分支和行波解(英文)[J]. 纺织高校基础科学学报,2013,26(3):6. TANG Yaning, CHEN Yanna. Bifurcations and traveling wave solutions for the(2+1)-dimensional Broer-Kaup-Kupershmidt equation[J]. Basic Sciences Journal of Textile Universities, 2013, 26(3):6.
[16] 李继彬.非线性波方程研究的动力系统方法和分支与混沌[EB/OL].(2007-01-29)[2022-1-10]. https://kns.cnki.net/kcms/detail/detail.aspx?FileName=SNAD000001222457&DbName=SNAD2012. LI Jibin. Nonlinear wave equations study on dynamical system methods and branching with chaos[EB/OL].(2007-01-29)[2022-1-10].https://kns.cnki.net/kcms/detail/detail.aspx?FileName=SNAD000001222457&DbName=SNAD2012.
[17] 李少勇.几类偏微分方程的非线性波解及其分支[D]. 广州:华南理工大学,2015. LI Shaoyong. Nonlinear wave solutions of several classes of partial differential equations and their branches[D]. Guangzhou: South China University of Technology, 2015.
[1] DONG Li. Lower bounds for blow up time of two nonlinear wave equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 56-60.
[2] FU Juan, ZHANG Rui, WANG Cai-jun, ZHANG Jing. The stability of a predator-prey diffusion model with Beddington-DeAngelis functional response [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(11): 115-122.
[3] DONG Jian-wei, LOU Guang-pu, WANG Yan-ping. Uniqueness of stationary solutions to a simplified energy-transport model for semiconductors [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 37-41.
[4] LÜ Hong-jie, LIU Jing-jing, QI Jing, LIU Shuo. Blow-up of a weakly dissipative μ-Hunter-Saxton equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(05): 55-59.
[5] DONG Jian-wei, CHENG Shao-hua, WANG Yan-ping. Classical solutions to stationary one-dimensional quantum energy-transport model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(03): 52-56.
[6] CHEN Zhi-hui, WANG Zhen-zhen, CHENG Yong-kuan*. Soliton solutions for quasilinear Schrdinger equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(2): 58-62.
[7] LIU Yan-qin,XU Ming-yu and JIANG Xiao-yun . The fractional nonlinear convection-diffusion equation and its solution [J]. J4, 2007, 42(1): 35-39 .
[8] SHI Chang-guang . Multi-soliton solution of the Faddeev model [J]. J4, 2007, 42(7): 38-40 .
[9] CHEN Peng-yu, MA Wei-feng, Ahmed Abdelmonem. Existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 13-23.
[10] ZHANG Shen-gui. Existence of solutions for fractional Kirchhoff-type equation with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 48-55.
[11] WANG Hai-quan, CHONG Ge-zi. Local Gevrey regularity and analyticity of the solutions to the initial value problem associated with the two-component Novikov system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 56-63.
[12] MENG Xi-wang, WANG Juan. Blow up of solutions to wave equations in the de Sitter spacetime [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 64-75.
[13] DU Ya-li, WANG Xuan. Pullback attractor for the non-autonomous Berger equation with nonlinear damping [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 30-41.
[14] ZHOU Yan-xia, WANG Xin-ru, XU Xiu-juan. Gradient estimates for weak solutions of A-harmonic equation under nonstandard growth [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 47-55.
[15] OUYANG Bai-ping, XIAO Sheng-zhong. Global nonexistence of solutions to a class of semilinear double-wave equations with space-dependent coefficients on the nonlinearity [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 59-65.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] YANG Jun. Characterization and structural control of metalbased nanomaterials[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 1 -22 .
[2] HE Hai-lun， CHEN Xiu-lan* . Circular dichroism detection of the effects of denaturants and buffers on the conformation of cold-adapted protease MCP-01 and  mesophilic protease BP01[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 23 -29 .
[3] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[4] SUN Xiao-ting1, JIN Lan2*. Application of DOSY in oligosaccharide mixture analysis[J]. J4, 2013, 48(1): 43 -45 .
[5] LUO Si-te, LU Li-qian, CUI Ruo-fei, ZHOU Wei-wei, LI Zeng-yong*. Monte-Carlo simulation of photons transmission at alcohol wavelength in  skin tissue and design of fiber optic probe[J]. J4, 2013, 48(1): 46 -50 .
[6] YANG Lun, XU Zheng-gang, WANG Hui*, CHEN Qi-mei, CHEN Wei, HU Yan-xia, SHI Yuan, ZHU Hong-lei, ZENG Yong-qing*. Silence of PID1 gene expression using RNA interference in C2C12 cell line[J]. J4, 2013, 48(1): 36 -42 .
[7] MAO Ai-qin1,2, YANG Ming-jun2, 3, YU Hai-yun2, ZHANG Pin1, PAN Ren-ming1*. Study on thermal decomposition mechanism of  pentafluoroethane fire extinguishing agent[J]. J4, 2013, 48(1): 51 -55 .
[8] YANG Ying, JIANG Long*, SUO Xin-li. Choquet integral representation of premium functional and related properties on capacity space[J]. J4, 2013, 48(1): 78 -82 .
[9] LI Yong-ming1, DING Li-wang2. The r-th moment consistency of estimators for a semi-parametric regression model for positively associated errors[J]. J4, 2013, 48(1): 83 -88 .
[10] DONG Wei-wei. A new method of DEA efficiency ranking for decision making units with independent subsystems[J]. J4, 2013, 48(1): 89 -92 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd