Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Abundant Solutions of Distinct Physical Structures for Three Shallow Water Waves Models

Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 295--304 | DOI:10.5890/DNC.2017.09.004


Department of Mathematics, Saint Xavier University, Chicago, IL 60655

Download Full Text PDF



In this work, we investigate three completely integrable model equations used to describe shallow water waves. A variety of techniques will be sued to determine abundant solutions, of distinct physical structures, for each model. The three models give soliton solutions, periodic solutions, rational hyperbolic functions and rational solutions as well.


  1. [1]  Wazwaz, A.M. (2008), The Hirotas direct method for multiple-soliton solutions for three model equations of shallow water waves, Appl. Math. Comput., 201, 489-503.
  2. [2]  Hereman, W. and Nuseir, A. (1997), Symbolic methods to construct exact solutions of nonlinear partial differential equations, Mathematics and Computers in Simulation, 43, 13-27.
  3. [3]  Hirota, R. (2004), The Direct Method in Soliton Theory, Cambridge University Press, Cambridge.
  4. [4]  Hirota, R. and Satsuma, J. (1976), N-soliton solutions of model equations for shallow water waves, J. Physical Society of Japan, 40(2), 611-612.
  5. [5]  Ito, M. (1980), An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher order, J. Physical Society of Japan, 49(2) 771-778.
  6. [6]  Sawada, K. and Kotera, T. (1974), A method for finding N-soliton solutions of the K.d.V. equation and K.d.V.-like equation, Prog. Theor. Phys., 511355-1367.
  7. [7]  Lax, P.D. (1968), Integrals of nonlinear equations of evolurion and solitary waves, Commun. Pure Appl. Math., 21, 467-490.
  8. [8]  Malfliet, W. (2004) The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, J. Computational and Applied Mathematics, (164-165), 529-541.
  9. [9]  Malfliet, W. and Hereman,W. (1996), The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54, 563-568.
  10. [10]  Malfliet, W. and Hereman, W. (1996), The Tanh Method: II. Perturbation technique for conservative systems, Physica Scripta, 54, 569-575.
  11. [11]  Wazwaz, A.M. (2002), PartialDifferential Equations:Methods and Applications, Balkema Publishers, The Netherlands.
  12. [12]  Wazwaz, A.M. (2015), Peakons and soliton solutions of newly Benjaniin-Bona-Mahony-like equations, Nonlinear Dynamics and Systems Theory, 15(2), 209-220.
  13. [13]  Wazwaz, A.M. (2012), Two kinds of multiple wave solutions for the potential YTSF equation and a potential YTSFtype equation, Journal of Applied Nonlinear Dynamics, 1(1), 51-58.
  14. [14]  Wazwaz, A.M. (2013), Multiple kink solutions for the (2+1)-dimensional Sharma-Tasso-Olver and the Sharma-Tasso- Olver-Burgers equations, Journal of Applied Nonlinear Dynamics, 2(1), 95-102.
  15. [15]  Wazwaz, A.M. (2014), Soliton solutions of the modified KdV6, modified (2+1)-dimensional Boussinesq equation, and (3+1)-dimensional KdV equation, Journal of Applied Nonlinear Dynamics, 3(1), 95-104.
  16. [16]  Wazwaz, A.M. (2012), One kink solution for a variety of nonlinear fifth-order equations, Discontinuity, Nonlinearity and Complexity, 1(2), 161-170.
  17. [17]  Wazwaz, A.M. (2007), The tanh-coth and the sech methods for exact solutions of the Jaulent-Miodek equation, Phys. Lett. A, 366(1/2), 85-90.