A New Simplified Bilinear Method for the N-Soliton Solutions for a Generalized FmKdV Equation with Time-Dependent Variable Coefficients

Marwan Alquran, H.M. Jaradat, Safwan Al-Shara’ and Fadi Awawdeh

Abstract

In this paper a generalized fractional modified Korteweg–de Vries (FmKdV) equation with time-dependent variable coefficients, which is a generalized model in nonlinear lattice, plasma physics and ocean dynamics, is investigated. With the aid of a simplified bilinear method, fractional transforms and symbolic computation, the corresponding N-soliton solutions are given and illustrated. The characteristic line method and graphical analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons and elastic interactions. Finally, the resonance phenomenon for the equation is examined.

  • [1]

    A. G. Nikitin and T. A. Barannyk, Solitary waves and other solutions for nonlinear heat equations, Cent. Eur. J. Math. 2 (2005), 840858.

    • Google Scholar
    • Export Citation
  • [2]

    A. Rafiq, M. Ahmed, and S. Hussain, A general approach to specific second order ordinary differential equations using homotopy perturbation method, Phys. Lett. A 372 (2008), 49734976.

    • Google Scholar
    • Export Citation
  • [3]

    V. S. Erturk, S. Momani, and Z. Odibat, Application of generalized differential transform method to multi-order fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 16421654.

    • Google Scholar
    • Export Citation
  • [4]

    V. Daftardar-Gejji and S. Bhalekar, Solving multi-term linear and non-linear diffusion wave equations of fractional order by a domain decomposition method, Appl. Math. Comput. 202 (2008), 113120.

    • Google Scholar
    • Export Citation
  • [5]

    V. Daftardar-Gejji and H. Jafari, Solving a multi-order fractional differential equation using a domain decomposition, Appl. Math. Comput. 189 (2007), 541548.

    • Google Scholar
    • Export Citation
  • [6]

    N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A 371 (2007), 2633.

    • Google Scholar
    • Export Citation
  • [7]

    A. Golbabai and K. Sayevand, Fractional calculus- A new approach to the analysis of generalized-fourth-order-diffusion wave equations, Comput. Math. Appl. 61 (2011), 22272231.

    • Google Scholar
    • Export Citation
  • [8]

    A. Golbabai and K. Sayevand, The homotopy perturbation method for multi-order time fractional differential equations, Nonlinear Sci Lett. A 1 (2010), 147154.

    • Google Scholar
    • Export Citation
  • [9]

    J. H. He, New interpretation of homotopy perturbation method, Int. J. Modern Phys. B 20 (2006), 17.

  • [10]

    S. Y. Lou and H. Y. Ruan, Conservation laws of the variable coefficient KdVS and mKdV equations, Acta Phys. Sin. 41 (1992), 182187.

    • Google Scholar
    • Export Citation
  • [11]

    W. Hong and Y. D. Jung, Auto-Bäcklund transformation and analytic solutions for general variable-coefficient KdV equation, Phys. Lett. A 257 (1999), 149152.

    • Google Scholar
    • Export Citation
  • [12]

    L. Juan, X. Tao, M. Xiang-Hua, Z. Ya-Xing, Z. Hai-Qiang, and T. Bo, Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation, J Math. Anal. Appl. 336 (2007), 14431455.

    • Google Scholar
    • Export Citation
  • [13]

    Y. Liu, Y. T. Gao, Z. Y. Sun, and X. Yu, Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves, Nonlinear Dyn. 66 (2011), 575587.

    • Google Scholar
    • Export Citation
  • [14]

    G. M. Wei, Y. T. Gao, W. Hu, and C. Y. Zhang, Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable coefficient Korteweg–de Vries (KdV) equation, Eur. Phys. J. B 53 (2006), 343350.

    • Google Scholar
    • Export Citation
  • [15]

    J. Weiss, M. Tabor, and G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys. 24 (1983), 522526.

    • Google Scholar
    • Export Citation
  • [16]

    M. Xiang-Hua, T. Bo, F. Qian, Y. Zhen-Zhi, and G. Yi-Tian, Painlevé analysis and determinant solutions of a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in Wronskian and Grammian form, Commun. Theor. Phys. 51 (2009), 10621068.

    • Google Scholar
    • Export Citation
  • [17]

    X. Yu, Y. T. Gao, Z. Y. Sun, and Y. Liu, N-soliton solutions, Bäcklund transformation and lax pair for a generalized variable-coefficient fifth-order Korteweg–de Vries equation, Phys. Scr. 81 (2010), 045402.

    • Google Scholar
    • Export Citation
  • [18]

    B. Tiana, G. M. Weic, C. Y. Zhange, and W. Rui, Transformations for a generalized variable-coefficient Korteweg–de Vries model from blood vessels, Bose–Einstein condensates, rods and positions with symbolic computation, Phys. Lett. A 356 (2006), 816.

    • Google Scholar
    • Export Citation
  • [19]

    F. Güng, M. Sanielevici, and P. Winternitz, On the integrability properties of variable coefficient Korteweg–de Vries equations, Can. J. Phys. 74 (1996), 676684.

    • Google Scholar
    • Export Citation
  • [20]

    R. Hirota, The direct method in soliton theory, Cambridge University Press, Cambridge, 2004.

  • [21]

    W. X. Ma, T. W. Huang, and Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Phys. Scripta. 82 (2010), 065003 (8pp).

    • Google Scholar
    • Export Citation
  • [22]

    W. X. Ma and E. G. Fan, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61 (2011), 950959.

    • Google Scholar
    • Export Citation
  • [23]

    W. X. Ma, A refined invariant subspace method and applications to evolution equations, Sci. China Math. 55 (2012), 17961778.

    • Google Scholar
    • Export Citation
  • [24]

    W. X. Ma, Z. Yi, T. Yaning, and J. Tu, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012), 71747183.

    • Google Scholar
    • Export Citation
  • [25]

    W. X. Ma, Bilinear equations and resonant solutions characterized by bell polynomials, Rep. Math. Phys. 72 (2013), 4156.

    • Google Scholar
    • Export Citation
  • [26]

    K. B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, 1974.

  • [27]

    G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable function: Further results, Comput. Math. Appl. 51 (2007), 13671376.

    • Google Scholar
    • Export Citation
  • [28]

    R. Hirota, Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 11921194.

    • Google Scholar
    • Export Citation
  • [29]

    R. Hirota, Direct methods in soliton theory, in: R. K. Bullough, P. J. Caudrey (Eds.), Solitons, Springer, Berlin, 1980.

    • Google Scholar
    • Export Citation
  • [30]

    R. Hirota, Exact N-soliton solutions of a nonlinear wave equation, J. Math. Phys. 14 (1973), 805809.

  • [31]

    R. Hirota and M. Ito , Resonance of solitons in one dimension, J. Phys. Soc. Jpn. 52 (1983), 744748.

  • [32]

    R. Hirota, Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (1972), 14561458.

    • Google Scholar
    • Export Citation
  • [33]

    W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, math, Comput. Simul. 43 (1997), 1327.

    • Google Scholar
    • Export Citation
  • [34]

    W. Hereman and W. Zhuang, A MACSYMA program for the Hirota method, 13th World Congress on Comput. Appl. Math. 2 (1991), 842863.

    • Google Scholar
    • Export Citation
  • [35]

    A. M. Wazwaz, Completely integrable coupled KdV and coupled KP systems, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 28282835.

    • Google Scholar
    • Export Citation
  • [36]

    A. M. Wazwaz, Multiple soliton solutions for a (2 + 1)-dimensional integrable KdV6 equation, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 14661472.

    • Google Scholar
    • Export Citation
  • [37]

    F. Awawdeh, H. M. Jaradat, and S. Al-Shara’, Applications of a simplified bilinear method to ion-acoustic solitary waves in plasma, Eur. Phys. J. D 66 (2012), 40.

    • Google Scholar
    • Export Citation
  • [38]

    H. M. Jaradat, S. Al-Shara’, F. Awawdeh, and M. Alquran, Variable coefficient equations of the Kadomtsev-Petviashvili hierarchy: Multiple soliton solutions and singular multiple soliton solutions, Phys. Scr. 85 (2012), 035001 (7pp).

    • Google Scholar
    • Export Citation
  • [39]

    F. Awawdeh, S. Al-Shara’, H. M. Jaradat, A. K. Alomari, and R. Alshorman, Symbolic computation on soliton solutions for variable-coefficient quantum Zakharov-Kuznetsov equation in magnetized dense plasmas, Int. J. Nonlinear Sci. Numer. Simulat. 15 (2014), 3545.

    • Google Scholar
    • Export Citation
  • [40]

    H. M. Jaradat, F. Awawdeh, S. Al-Shara’, M. Alquran, and S. Momani, Controllable dynamical behaviors and the analysis of fractal burgers hierarchy with the full effects of inhomogeneities of media, Romanian J. Phys. 60 (3–4) (2015), 324–343.

    • Google Scholar
    • Export Citation
  • [41]

    V. E. Tarasov, No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci. Numer. Simulat. 18 (2013), 29452948.

    • Google Scholar
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Price including VAT
Log in
Already have access? Please log in.


Journal + Issues

The International Journal of Nonlinear Sciences and Numerical Simulation publishes original papers on all subjects relevant to nonlinear sciences and numerical simulation. The journal is directed at researchers in nonlinear sciences, engineers, and computational scientists, economists, and others, who either study the nature of nonlinear problems or conduct numerical simulations of nonlinear problems.

Search