Skip to main content
Log in

New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In the present course of study, we examine a family of Boussinesq equations of distinct structures and dimensions. We investigate the complete integrability of these equations via Painlevé test. Real and complex multiple soliton solutions, for each considered model, are derived by mode of simplified Hirota’s method. Moreover, exponential expansion method has been employed to each equation, resulting into soliton solutions possessing rich spatial structure due to the presence of abundant arbitrary constants.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boussinesq, J.V.: Essai sur la théorie des eaux courantes. Mm. Prsents Divers Savants Acad. Sci. Inst. Nat. Fr. XXIII, 55–108 (1877)

    MATH  Google Scholar 

  2. Darvishi, M., Najafi, M., Wazwaz, A.M.: Soliton solutions for Boussinesq-like equations with spatio-temporal dispersion. Ocean Eng. 130, 228–240 (2017)

    Article  Google Scholar 

  3. McKean, H.P.: Boussinesq’s equation as a Hamiltonian system. Adv. Math. Supp. Studies 3, 217–226 (1978)

    MathSciNet  MATH  Google Scholar 

  4. McKean, H.P.: Boussinesq’s equation on the circle. Commun. Pure Appl. Math. 34, 599–691 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clarkson, P.A., Kruskal, M.D.: New similarity solutions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhu, J.Y.: Line-soliton and rational solutions to (2+1)-dimensional Boussinesq equation by Dbar-problem (2017). arXiv:1704.02779v2

  7. Hirota, R., Ito, M.: Resonance of solitons in one dimension. J. Physical Soc. Japan 52, 744–748 (1983)

    Article  MathSciNet  Google Scholar 

  8. Hereman, W., Nuseir, A.: Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simul. 43, 13–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  10. Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer and HEP, Berlin (2009)

    Book  MATH  Google Scholar 

  11. Wazwaz, A.M.: Two kinds of multiple wave solutions for the potential YTSF equation and a potential YTSF-type equation. J. Appl. Nonlinear Dyn. 1, 51–58 (2012)

    Article  MATH  Google Scholar 

  12. Leblond, H., Mihalache, D.: Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Adem, A.R., Khalique, C.M.: New exact solutions and conservation laws of a coupled Kadomtsev–Petviashvili system. Comput. Fluids 81, 10–16 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wazwaz, A.M.: Multiple kink solutions for the (2+1)-dimensional Sharma–Tasso–Olver and the Sharma–Tasso–Olver–Burgers equations. J. Appl. Nonlinear Dyn. 2, 95–102 (2013)

    Article  MATH  Google Scholar 

  15. Su, T.: Explicit solutions for a modified (2+1)-dimensional coupled Burgers equation by using Darboux transformation. Appl. Math. Lett. 69, 15–21 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mihalache, D.: Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature. Rom. Rep. Phys. 69, 403–420 (2017)

    Google Scholar 

  17. Xing, Q., Wu, Z., Mihalache, D., He, Y.: Smooth positon solutions of the focusing modified Korteweg-de Vries equation. Nonlinear Dyn. 89, 2299–2310 (2017)

    Article  MathSciNet  Google Scholar 

  18. Wazwaz, A.M.: One kink solution for a variety of nonlinear fifth-order equations. Discontin. Nonlinearity Complex. 1, 161–170 (2012)

    Article  MATH  Google Scholar 

  19. Wazwaz, A.M.: Abundant solutions of distinct physical structures for three shallow water waves models. Discontin. Nonlinearity Complex. 6, 295–304 (2017)

    Article  Google Scholar 

  20. Wazwaz, A.M.: A variety of distinct kinds of multiple soliton solutions for a (3+1)-dimensional nonlinear evolution equations. Math. Methods Appl. Sci. 36, 349–357 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wazwaz, A.M.: Multiple real and multiple complex soliton solutions for the integrable Sine–Gordon equation. Optik 172, 622–627 (2018)

    Article  Google Scholar 

  22. 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)

    Article  Google Scholar 

  23. Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property of partial differential equations. J. Math. Phys. A 24, 522–526 (1983)

    Article  MATH  Google Scholar 

  24. Kaur, L., Wazwaz, A.M.: Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation. Nonlinear Dyn. 94, 2469–2477 (2018)

    Article  Google Scholar 

  25. Kaur, L., Wazwaz, A.M.: Einstein’s vacuum field equation: Painlevé analysis and Lie symmetries. Waves Random Complex (2019). https://doi.org/10.1080/17455030.2019.1574410. in press

    Google Scholar 

  26. Yin, Y.H., Ma, W.X., Liu, J.G., Lu, X.: Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction. Comput. Math. Appl. 76, 1275–1283 (2018)

    Article  MathSciNet  Google Scholar 

  27. Gao, L.N., Zi, N.N., Yin, Y.H., Ma, W.X., Lu, X.: B\(\ddot{a}\)cklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 89, 2233–2240 (2017)

    Article  Google Scholar 

  28. Lu, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lu, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 86, 523–534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lu, X., Ma, W.X., Khalique, C.M.: A direct bilinear Backlund transformation of a (2+1)-dimensional Korteweg-de Vries-like model. Appl. Math. Lett. 50, 37–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lu, X., Ma, W.X., Yum, J., Khalique, C.M.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 31(1–3), 40–46 (2016)

    Article  MathSciNet  Google Scholar 

  32. Clarkson, P., Dowie, E.: Rational solutions of the Boussinesq equation and applications to rogue waves. Trans. Math. Appl. 1(1), tnx003 (2017). https://doi.org/10.1093/imatrm/tnx003

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abdul-Majid Wazwaz.

Ethics declarations

Conflict of interest

The authors declare they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wazwaz, AM., Kaur, L. New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions. Nonlinear Dyn 97, 83–94 (2019). https://doi.org/10.1007/s11071-019-04955-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04955-1

Keywords

Navigation