Nonlinear Dynamics

, Volume 94, Issue 4, pp 2655–2663 | Cite as

Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions

  • Abdul-Majid WazwazEmail author
Original Paper


In this paper, we develop two new fourth-order integrable equations represented by nonlinear PDEs of second-order derivative in time t. The new equations model both right- and left-going waves in a like manner to the Boussinesq equation. We will employ the Painlevé analysis to formally show the complete integrability of each equation. The simplified Hirota’s method is used to derive multiple soliton solutions for this equation. We introduce a complex form of the simplified Hirota’s method to develop multiple complex soliton solutions. More exact traveling wave solutions for each equation will be derived as well.


Fourth-order integrable equation Painlevé test Multiple soliton solutions Multiple complex soliton solutions 


Compliance with ethical standards

Conflicts of interest

The author declares no conflict of interest.


  1. 1.
    Wazwaz, A.M.: A new fifth order nonlinear integrable equation: multiple soliton solutions. Physica Scripta 83, 015012 (2011)CrossRefGoogle Scholar
  2. 2.
    Wazwaz, A.M.: A new generalized fifth-order nonlinear integrable equation. Phys. Scr. 83, 035003 (2011)CrossRefGoogle Scholar
  3. 3.
    Osman, M.S., Machado, J.A.T.: New nonautonomous combined multi-wave solutions for (2+1)-dimensional variable coefficients KdV equation. Nonlinear Dyn. (2018). (In Press). CrossRefGoogle Scholar
  4. 4.
    Osman, M.S., Machado, J.A.T.: The dynamical behavior of mixed type soliton solutions described by (2+1)-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients. J. Electromagn. Waves Appl. 32(11), 1457–1464 (2018)CrossRefGoogle Scholar
  5. 5.
    Baldwin, D., Hereman, W.: Symbolic software for the Painlevé test of nonlinear ordinary and partial differential equations. J. Nonlinear Math. Phys. 13(1), 90–110 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Verheest, F., Olivier, C.P., Hereman, W.: Modified Korteweg-de Vries solitons at supercritical densities in two-electron temperature plasmas. J. Plasma Phys. 82(02), 905820208 (2016)CrossRefGoogle Scholar
  7. 7.
    Fokas, A.: Symmetries and integrability. Stud. Appl. Math. 77, 253–299 (1987)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hirota, R.: A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog. Theor. Phys. 52(5), 1498–1512 (1974)CrossRefGoogle Scholar
  9. 9.
    Sanders, J., Wang, P.: Integrable systems and their recursion operators. Nonlinear Anal. 47, 5213–5240 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Magri, F.: Lectures Notes in Physics. Springer, Berlin (1980)Google Scholar
  11. 11.
    Baldwin, D., Hereman, W.: A symbolic algorithm for computing recursion operators of nonlinear partial differential equations. Int. J. Comput. Math. 87(5), 1094–1119 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Poole, D., Hereman, W.: Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions. J. Symb. Comput. 46(12), 1355–1377 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khoury, S.A.: New anstz for obtaining wave solutions of the generalized CamassaHolm equation. Chaos Solitons Fractals 25(3), 705–710 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Leblond, H., Mihalache, D.: Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Leblond, H., Mihalache, D.: Few-optical-cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models. Phys. Rev. A 79, 063835 (2009)CrossRefGoogle Scholar
  16. 16.
    Khalique, C.M.: Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities. Pramana 80, 413–427 (2013)CrossRefGoogle Scholar
  17. 17.
    Kara, A.H., Khalique, C.M.: Nonlinear evolution-type equations and their exact solutions using inverse variational methods. J. Phys. A Math. Gen. 38, 4629–4636 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theorem. Springer, Berlin (2009)CrossRefGoogle Scholar
  19. 19.
    Wazwaz, A.M.: \(N\)-soliton solutions for the Vakhnenko equation and its generalized forms. Phys. Scr. 82, 065006 (2010)CrossRefGoogle Scholar
  20. 20.
    Wazwaz, A.M.: Multiple soliton solutions for the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation. Nonlinear Anal. Ser. A Theory Methods Appl. 72, 1314–1318 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wazwaz, A.M.: Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation. Open Eng. 7, 169–174 (2017)CrossRefGoogle Scholar
  22. 22.
    Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83(1), 591–596 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wazwaz, A.M.: Two wave mode higher-order modified KdV equations: essential conditions for multiple soliton solutions to exist. J. Numer. Methods Heat Fluid Flow 27(10), 2223–2230 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA

Personalised recommendations