Research article

Exact explicit nonlinear wave solutions to a modified cKdV equation

  • Received: 03 December 2019 Accepted: 31 May 2020 Published: 03 June 2020
  • MSC : 34C60, 35B3, 35C07

  • In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.

    Citation: Zhenshu Wen, Lijuan Shi. Exact explicit nonlinear wave solutions to a modified cKdV equation[J]. AIMS Mathematics, 2020, 5(5): 4917-4930. doi: 10.3934/math.2020314

    Related Papers:

  • In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.


    加载中


    [1] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc., London, 1991.
    [2] Z. Zhao, Y. Xu, Solitary waves for Korteweg-de Vries equation with small delay, J. Math. Anal. Appl., 368 (2010), 43-53.
    [3] J. Liu, J. Guan, Z. Feng, Hopf bifurcation analysis of KdV-Burgers-Kuramoto chaotic system with distributed delay feedback, Int. J. Bifurcat. Chaos, 29 (2019), 1-13.
    [4] L. Baudouin, E. Crepeau, J. Valein, Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback, IEEE T. Automat. Contr., 64 (2019), 1403-1414.
    [5] V. Komornik, C. Pignotti, Well-posedness and exponential decay estimates for a Korteweg-de Vries-Burgers equation with time-delay, Nonlinear Anal. Theor., 191 (2020), 1-13.
    [6] M. S. Alber, G. G. Luther, C. A. Miller, On soliton-type solutions of equations associated with N-component systems, J. Math. Phys., 41 (2000), 284-316.
    [7] M. Antonowicz, A. P. Fordy, Coupled KdV equations with multi-Hamiltonian structures, Physica D, 28 (1987), 345-357.
    [8] M. Antonowicz, A. P. Fordy, A family of completely integrable multi-Hamiltonian systems, Phys. Lett. A, 122 (1987), 95-99.
    [9] M. S. Alber, G. G. Luther, J. E. Marsden, Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces, Nonlinearity, 10 (1997), 1-24.
    [10] Z. Wen, Q. Wang, Abundant exact explicit solutions to a modified cKdV equation, J. Nonlinear Model. Anal., 1 (2020), 1-12.
    [11] Z. Wen, Z. Liu, M. Song, New exact solutions for the classical Drinfel'd-Sokolov-Wilson equation, Appl. Math. Comput., 215 (2009), 2349-2358.
    [12] Z. Wen, Qualitative study of effects of vorticity on traveling wave solutions to the two-component Zakharovcit system, Appl. Anal., (2019), 1250305.
    [13] J. Li, Z. Qiao, Bifurcations and exact traveling wave solutions of the generalized two-component Camassa-Holm equation, Int. J. Bifurcat. Chaos, 22 (2012), 1-13.
    [14] Z. Wen, Bifurcation of solitons, peakons, and periodic cusp waves for θ-equation, Nonlinear Dynam., 77 (2014), 247-253.
    [15] Z. Wen, L. Shi, Dynamics of bounded traveling wave solutions for the modified Novikov equation, Dynam. Syst. Appl., 27 (2018), 581-591.
    [16] A. Biswas, M. Song, Soliton solution and bifurcation analysis of the Zakharov-Kuznetsov-Benjamin-Bona-Mahoney equation with power law nonlinearity, Commun. Nonlinear Sci., 18 (2013), 1676-1683.
    [17] Z. Wen, Several new types of bounded wave solutions for the generalized two-component Camassa-Holm equation, Nonlinear Dynam., 77 (2014), 849-857.
    [18] Z. Wen, Bifurcations and nonlinear wave solutions for the generalized two-component integrable Dullin-Gottwald-Holm system, Nonlinear Dynam., 82 (2015), 767-781.
    [19] C. Pan, Y. Yi, Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity, J. Nonlinear Math. Phys., 22 (2015), 308-320.
    [20] Z. Wen, Bifurcations and exact traveling wave solutions of a new two-component system, Nonlinear Dynam., 87 (2017), 1917-1922.
    [21] Z. Wen, Bifurcations and exact traveling wave solutions of the celebrated Green-Naghdi equations, Int. J. Bifurcat. Chaos, 27 (2017), 1-7.
    [22] T. D. Leta, J. Li, Various exact soliton solutions and bifurcations of a generalized Dullin-Gottwald-Holm equation with a power law nonlinearity, Int. J. Bifurcat. Chaos, 27 (2017), 1-22.
    [23] L. Shi, Z. Wen, Bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation, Commun. Theor. Phys., 69 (2018), 631-636.
    [24] Z. Wen, Abundant dynamical behaviors of bounded traveling wave solutions to generalized θ-equation, Comp. Math. Math. Phys., 59 (2019), 926-935.
    [25] L. Shi, Z. Wen, Dynamics of traveling wave solutions to a highly nonlinear Fujimoto-Watanabe equation, Chinese Phys. B, 28 (2019), 1-7.
    [26] A. R. Seadawy, D. Lu, M. M. Khater, Bifurcations of traveling wave solutions for Dodd-Bullough-Mikhailov equation and coupled Higgs equation and their applications, Chinese J. Phys., 55 (2017), 1310-1318.
    [27] L. Shi, Z. Wen, Several types of periodic wave solutions and their relations of a Fujimoto-Watanabe equation, J. Appl. Anal. Comput., 9 (2019), 1193-1203.
    [28] Z. Wen, The generalized bifurcation method for deriving exact solutions of nonlinear space-time fractional partial differential equations, Appl. Math. Comput., 366 (2020), 1-10.
    [29] P. Byrd, M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, Berlin, 1971.
    [30] Z. Wen, On existence of kink and antikink wave solutions of singularly perturbed Gardner equation, Math. Method. Appl. Sci., 43 (2020), 4422-4427.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3379) PDF downloads(254) Cited by(2)

Article outline

Figures and Tables

Figures(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog