2021 Volume 11 Issue 1
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Shaolong Xie, Xiaochun Hong, Junliang Lu. THE BIFURCATION TRAVELLING WAVES OF A GENERALIZED BROER-KAUP EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 210-226. doi: 10.11948/20190268
Citation: Shaolong Xie, Xiaochun Hong, Junliang Lu. THE BIFURCATION TRAVELLING WAVES OF A GENERALIZED BROER-KAUP EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 210-226. doi: 10.11948/20190268

THE BIFURCATION TRAVELLING WAVES OF A GENERALIZED BROER-KAUP EQUATION

  • Corresponding author: Email address: xchong@ynufe.edu.cn(X. Hong) 
  • Fund Project: The authors were supported by National Natural Science Foundation of China(11761075)
  • In this paper, by using the dynamical system theory and simulation method, we study a generalized Broer-Kaup (gBK) equation. Under different parameter conditions, the travelling wave solutions such as kink wave, periodic blow-up wave, periodic wave and solitary wave are given, and the dynamic characters of these solutions are investigated.
    MSC: 35B10, 35C07, 35C08
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  • [1] A. E. Achab and A. Amine, A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg-Landau equation, Nonlinear Dynamics, 2018, 91(2), 995-999. doi: 10.1007/s11071-017-3924-0

    CrossRef Google Scholar

    [2] D. Andrade, R. Stuhlmeier and M. Stiassnie, On the generalized kinetic equation for surface gravity waves, blow-up and its restraint, Fluids, 2019, 4(2), doi:10.3390/fluids4010002, 1-18.

    CrossRef Google Scholar

    [3] M. Arshad, A. R. Seadawy, D. Lu and J. Wang, Travelling wave solutions of Drinfel'd-Sokolov-Wilson, Whitham-Broer-Kaup and (2+1)-dimensional Broer-Kaup-Kupershmit equations and their applications, Chinese Journal of Physics, 2017, 55(3), 780-797. doi: 10.1016/j.cjph.2017.02.008

    CrossRef Google Scholar

    [4] P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and scientists, Berlin, Springer, 1971.

    Google Scholar

    [5] A. Dur´ an, An efficient method to compute solitary wave solutions of fractional Korteweg-de Vries equations, International Journal of Computer Mathematics, 2018, 95(6-7), 1362-1374. doi: 10.1080/00207160.2017.1422732

    CrossRef Google Scholar

    [6] E. Fan, A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equation, J. Math. Phys., 2001, 42(9), 4327-4344. doi: 10.1063/1.1389288

    CrossRef Google Scholar

    [7] E. Fan, Integrable evolution systems based on Gerdjikov-Ivanov equations, biHamiltonian strutcure, finiti-dimensional integrable system s and N-fold Darboux transformation, J. Math. Phys., 2000, 41(11), 7769-7782. doi: 10.1063/1.1314895

    CrossRef Google Scholar

    [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Sixth ed., Academic Press, 2000.

    Google Scholar

    [9] X. Hu, A powerful approach to generate new integrable systems, J. Phys. A, 1994, 27(7), 2497-2514. doi: 10.1088/0305-4470/27/7/026

    CrossRef Google Scholar

    [10] T. D. Leta, A. El Achab, W. Liu and J. Ding, Application of bifurcation method and rational sine-Gordon expansion method for solving 2D complex Ginzburg-Landau equation, International Journal of Modern Physics B, 2020, 34(9), 2050079, 1-19.

    Google Scholar

    [11] J. Li, Geometric properties and exact travelling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation, Journal of Nonlinear Modeling and Analysis, 2019, 1(1), 1-10.

    Google Scholar

    [12] J. Liang and J. Li, Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity, J. Appl. Anal. Comput., 2018, 8(4), 1194-1210.

    Google Scholar

    [13] W. Ma, A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction, Chin. J. Contemp. Math., 1992, 13(1), 79-89.

    Google Scholar

    [14] W. Ma, The algebraic structure related to L-A-B triad representations of integrable system, Chin. Sci. Bull., 1992, 37(15), 1249-1253.

    Google Scholar

    [15] W. Ma, The Lie algebra structures of time-dependent symmetries of evolution equations, Acta Math. Appl. Sinica, 1994, 17(3), 388-392.

    Google Scholar

    [16] J. A. Pava, Stability properties of solitary waves for fractional KdV and BBM equations, Nonlinearity, 2018, 31(3), 920-956. doi: 10.1088/1361-6544/aa99a2

    CrossRef Google Scholar

    [17] G. Tu, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 1989, 39(2), 330-338.

    Google Scholar

    [18] S. Xie and J. Cai, Exact compacton and generalized kink wave solutions of the extended reduced Ostrovsky equation, Commun Nonlinear Sci Numer Simulat., 2009, 14(9-10), 3561-3573. doi: 10.1016/j.cnsns.2009.02.001

    CrossRef Google Scholar

    [19] S. Xie, X. Hong and T. Jiang, Planar bifurcation method of dynamical system for investigating different kinds of bounded travelling wave solutions of a generalized Camassa-Holm equation, J. Appl. Anal. Comput., 2017, 7(1), 278-290.

    Google Scholar

    [20] S. Xie, Q. Lin and B. Gao, Periodic and solitary travelling-wave solutions of a CH-DP equation, Commun Nonlinear Sci Numer Simul, 2011, 16(11), 3941- 3948.

    Google Scholar

    [21] S. Xie and L. Wang, Compacton and generalized kink wave solutions of the CH-DP equation, Appl. Math. Comput., 2010, 215(11), 4028-4039.

    Google Scholar

    [22] S. Xie, L. Wang and Y. Zhang, Explicit and implicit solutions of a generalized Camassa-Holm Kadomtsev-Petviashvili equation, Commun Nonlinear Sci Numer Simu, 2012, 17(3), 1130-1141. doi: 10.1016/j.cnsns.2011.07.003

    CrossRef Google Scholar

    [23] Y. Zhang, Z. Han and H. Tam, An integrable hierarchy and Darboux transformations, bilinear Backlund transformations of a reduced equation, Appl. Math. Comput., 2013, 219(11), 5837-5848.

    Google Scholar

    [24] Y. Zhang and H. Zhang, A method for integrable coupling of TD hierarchy, J. Math. Phys., 2002, 43(1), 466-472. doi: 10.1063/1.1398061

    CrossRef Google Scholar

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