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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Algebraic Traveling Wave Solutions to Nonlinear Evolution Equations

Journal of Applied Nonlinear Dynamics 8(4) (2019) 557--567 | DOI:10.5890/JAND.2019.12.004

Yakup Yıldırım$^{1}$, Emrullah Yaşar$^{2}$

$^{1}$ Department of Mathematics, Faculty of Arts and Sciences, Near East University, 99138 Nicosia, Cyprus

$^{2}$ Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059, Bursa, Turkey

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In this paper, we employ planar dynamical systems and invariant algebraic curves to characterize all algebraic traveling wave solutions to nonlinear evolution equations. In order to demonstrate the applicability and efficiency of the method, we apply the approach to four (2+1)-dimensional integrable extensions of the Kadomtsev–Petviashvili equation. The numerical simulations are also plotted for better understanding the physical phenomena.


  1. [1]  Ablowitz, M.J., Ablowitz, M.A., Clarkson, P.A., and Clarkson, P.A. (1991), Solitons, nonlinear evolution equations and inverse scattering, Cambridge university press: Cambridge.
  2. [2]  Vakhnenko, V.O., Parkes, E.J., and Morrison, A.J. (2003), A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation, Chaos, Solitons & Fractals, 17, 683- 692.
  3. [3]  Hirota, R. (1976), Direct method of finding exact solutions of nonlinear evolution equations, Springer: Berlin.
  4. [4]  Malfliet, W. and Hereman, W. (1996), The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54, 563.
  5. [5]  Wazwaz, A.M. (2004), The tanh method for traveling wave solutions of nonlinear equations, Applied Mathematics and Computation, 154, 713-723.
  6. [6]  Tariq, H. and Akram, G. (2017), New traveling wave exact and approximate solutions for the nonlinear Cahn–Allen equation: evolution of a nonconserved quantity, Nonlinear Dynamics, 88, 581-594.
  7. [7]  Ali, S., Rizvi, S.T.R., and Younis, M. (2015), Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients, Nonlinear Dynamics, 82, 1755-1762.
  8. [8]  Rizvi, S.T.R. and Ali, K. (2017), Jacobian elliptic periodic traveling wave solutions in the negative-index materials, Nonlinear Dynamics, 87, 1967-1972.
  9. [9]  Wazwaz, A.M. (2016), Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations, Nonlinear Dynamics, 85, 731-737.
  10. [10]  Wazwaz, A.M. and El-Tantawy, S.A. (2016), A new integrable (3+1)-dimensional KdV-like model with its multiple-soliton solutions, Nonlinear Dynamics, 83, 1529-1534.
  11. [11]  Ma, W.X., Huang, T., and Zhang, Y. (2010), A multiple exp-function method for nonlinear differential equations and its application, Physica Script a, 82, 065003.
  12. [12]  He, J.H. andWu, X.H. (2006), Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30, 700-708.
  13. [13]  Feng, Z. (2002), The first-integral method to study the Burgers–Korteweg–de Vries equation, Journal of Physics A: Mathematical and General, 35, 343.
  14. [14]  Ma, W.X. (2013), Bilinear equations, Bell polynomials and linear superposition principle, In Journal of Physics: Conference Series, 411, 012021.
  15. [15]  Gasull, A. and Giacomini, H. (2015), Explicit travelling waves and invariant algebraic curves, Nonlinearity, 28, 1597.
  16. [16]  Valls, C. (2017), Complete characterization of algebraic traveling wave solutions for the Boussinesq, Klein– Gordon and Benjamin–Bona–Mahony equations, Chaos, Solitons & Fractals, 95, 148-151.
  17. [17]  Chen, A., Zhu, W., Qiao, Z., and Huang, W. (2014), Algebraic traveling wave solutions of a non-local hydrodynamic-type model, Mathematical Physics, Analysis and Geometry, 17, 465-482.
  18. [18]  Grimshaw, R. (2007), Nonlinear waves in fluids: recent advances and modern applications, Springer: New York.
  19. [19]  Biondini, G. and Pelinovsky, D. (2008), Kadomtsev-Petviashvili equation, Scholarpedia, 3, 6539.
  20. [20]  Wazwaz, A.M. (2010), Four (2+1)-dimensional integrable extensions of the Kadomtsev–Petviashvili equation, Applied Mathematics and Computation, 215, 3631-3644.