Skip Navigation Links
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

Fax: +1 618 650 2555 Email:

New (2+1) and (3+1)-dimensional Integrable Boussinesq Equations: Multiple Soliton Solutions

Journal of Applied Nonlinear Dynamics 8(4) (2019) 549--556 | DOI:10.5890/JAND.2019.12.003

Abdul-Majid Wazwaz

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

Download Full Text PDF



In this work, we investigate new (2+1) and (3+1)-dimensional Boussinesq equations. We show the complete integrability of these equations via using the Painlevé test. We derive multiple soliton solutions for each model by using the simplified Hirota’s method. Other exact solutions of physical structures are determined.


  1. [1]  Fokas, A. (1987), Symmetries and integrability, Studies in Applied Mathematics, 77, 253-299.
  2. [2]  Guthrie, G. (1994), Recursion operators and non-local symmetries, Proceedings: Mathematical and Physical Sciences, 446, 107-114.
  3. [3]  Olver, P.J. (1977), Evolution equations possessing infinitely many symmetries, Journal of Mathematical Physics, 18(6), 1212-1215.
  4. [4]  Wazwaz, A.M. (2013), A variety of distinct kinds of multiple soliton solutions for a (3+1)-dimensional nonlinear evolution equations, Mathematical Methods in Applied Sciences, 36, 349-357.
  5. [5]  Weiss, J., Tabor, M., and Carnevale, G. (1983), The Painlev´e property of partial differential equations, J. Math. Phys. A, 24, 522-526.
  6. [6]  Hirota, R. and Ito, M. (1983), Resonance of solitons in one dimension, J. Phys. Soc. Japan, 52(3), 744-748.
  7. [7]  Zhu, J.Y. (2017), Line-soliton and rational solutions to (2+1)-dimensional Boussinesq equation by Dbarproblem, arXiv:1704.02779v2.
  8. [8]  Su, T. (2017), Explicit solutions for a modified 2+1-dimensional coupled Burgers equation by using Darboux transformation, Appl. Math. Lett., 69, 15-21.
  9. [9]  Mihalache, D. (2017), Multidimensional localized structures in optical and matter-wave media: A topical survey of recent literature, Rom. Rep. Phys., 69, 403.
  10. [10]  Xing, Q., Wu, Z., Mihalache, D., and He, Y. (2017), Smooth positon solutions of the focusing modi fied Korteweg-de Vries equation, Nonl. Dyn., 89, 2299-2310.
  11. [11]  Leblond, H. and Mihalache, D. (2013), Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep., 523, 61-126.
  12. [12]  McKean, H.P. (1978), Boussinesq’s equation as a Hamiltonian system, Adv. Math. Supp. Studies, 3, 217-226.
  13. [13]  McKean, H.P. (1981), Boussinesq’s equation on the circle, Commun. Pure Appl. Math., 34, 599-691.
  14. [14]  Clarkson, P. A. and Kruskal, M.D. (1989), New similarity solutions of the Boussinesq equation, J. Math. Phys., 30, 2201-2213.
  15. [15]  Hirota, R. (2004), The Direct Method in Soliton Theory, Cambridge University Press, Cambridge.
  16. [16]  Adem, A.R. and Khalique, C.M. (2013), New exact solutions and conservation laws of a coupled Kadomtsev- Petviashvili system, Computers & Fluids, 81, 10-16.
  17. [17]  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.
  18. [18]  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.
  19. [19]  Wazwaz, A.M. (2012), Two kinds of multiple wave solutions for the potential YTSF equation and a potential YTSF-type equation, Journal of Applied Nonlinear Dynamics, 1(1), 51-58.
  20. [20]  Wazwaz, A.M. (2009), Partial Differential Equations and Solitary Waves Theory, Springer and HEP, Berlin.
  21. [21]  Wazwaz, A.M. (2017), Abundant solutions of distinct physical structures for three shallow water waves models, Discontinuity, Nonlinearity, and Complexity, 6(3), 295-304.
  22. [22]  Wazwaz, A.M. (2012), One kink solution for a variety of nonlinear fifth-order equations, Discontinuity, Nonlinearity and Complexity, 1(2), 161-170.
  23. [23]  Darvishi, M., Najafi, M., and Wazwaz, A.M. (2017), Soliton solutions for Boussinesq-like equations with spatio-temporal dispersion, Ocean Engineering, 130, 228-240.