Skip to main content
SpringerLink
Log in

New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada–Kotera model

Nonlinear Dynamics volume 76, pages 161–168 (2014)Cite this article

Abstract

In this paper, the (2+1)-dimensional Sawada–Kotera model is studied with the Hirota bilinear method, gauge transformation and symbolic computation. Based on an alternative bilinear representation of the model, a bilinear Bäcklund transformation (BT) with three arbitrary constants is derived. Via applying a gauge transformation to this BT and choosing suitable constant parameters, three other sets of bilinear BTs are constructed, among which, the last set is treated as a new bilinear BT and denoted as BTIV hereby. Finally, by performing the perturbation technique on the new bilinear BT, namely BTIV, multisoliton solutions are iteratively achieved, and as an example, the one-, two- and three-soliton solutions are explicitly given. Note that formulas of the soliton solutions obtained hereby through solving the BTIV are different from the previous ones in other literature.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Zhang, Y., Song, Y., Cheng, L., Ge, J.Y., Wei, W.W.: Exact solutions and Painlevé analysis of a new (2+1)-dimensional generalized KdV equation. Nonlinear Dyn. 68, 445 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Lü, X., Tian, B., Zhang, H.Q., Xu, T., Li, H.: Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäklund transformation, Lax representation and interaction mechanisms. Nonlinear Dyn. 67, 2279 (2012)

    Article  MATH  Google Scholar 

  3. Hu, X.B.: Nonlinear superposition formulae for the differential-difference analogue of the KdV equation and two-dimensional Toda equation. J. Phys. A 27, 201 (1994)

    Article  MATH  Google Scholar 

  4. Dai, C.Q., Wang, Y.Y., Tian, Q., Zhang, J.F.: The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation. Ann. Phys. 327, 512 (2012)

    Article  MATH  Google Scholar 

  5. Hu, X.B., Li, Y.: Nonlinear superposition formula of 1+2-dimensional Caudrey–Dodd–Dibbon–Kotera–Sawada equation. Appl. Math. J. Chin. Univ. Ser. A 8, 17 (1993)

    Google Scholar 

  6. Ma, W.X., Fan, E.G.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61, 950 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ma, W.X.: Diversity of exact solutions to a restricted Boiti–Leon–Pempinelli dispersive long-wave system. Phys. Lett. A 319, 325 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ma, W.X., Wu, H.Y., He, J.S.: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A 364, 29 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Xu, X.X.: A family of integrable differential-difference equations, its bi-Hamiltonian structure and binary nonlinearization of the Lax pairs and adjoint Lax pairs. Chaos Solitons Fractals 45, 444 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lü, X., Peng, M.: Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications. Nonlinear Dyn. 73, 405 (2013)

    Article  MATH  Google Scholar 

  11. Lü, X., Peng, M.: Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells–Fokas model. Chaos 23, 013122 (2013)

    Article  Google Scholar 

  12. Lü, X., Peng, M.: Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics. Commun. Nonlinear Sci. Numer. Simul. 18, 2304 (2013)

    Article  MathSciNet  Google Scholar 

  13. Lv, N., Mei, J.Q., Zhang, H.Q.: Differential form method for finding symmetries of a (2+1)-dimensional Camassa–Holm system based on its Lax pair. Chaos Solitons Fractals 45, 503 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Olver, P.J.: Applications of Lie Groups to Differential Equations: Lecture Notes. Oxford University Mathematical Institute, Oxford (1980)

    MATH  Google Scholar 

  15. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  16. Konopelchenko, B.G.: Solitons in Multidimensions. World Scientific, Singapore (1993)

    Book  MATH  Google Scholar 

  17. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  19. Rogers, C., Shadwick, W.F.: Bäcklund Transformations and Their Applications. Academic Press, New York (1982)

    MATH  Google Scholar 

  20. Zhang, C.Y., Gao, Y.T., Xu, T., Li, L.L., Sun, F.W., Li, J., Meng, X.H., Wei, G.M.: Various methods for constructing auto-Bäcklund transformations for a generalized variable-coefficient Korteweg–de Vries model from plasmas and fluid dynamics. Commun. Theor. Phys. 49, 673 (2008)

    Article  MathSciNet  Google Scholar 

  21. Wadati, M., Sanuki, H., Konno, K.: Relationship among inverse method, Bäcklund transformation and an infinite number of conservation laws. Prog. Theor. Phys. 53, 419 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hirota, R.: Soliton solutions to the BKP equations. II. The integral equation. J. Phys. Soc. Jpn. 58, 2705 (1989)

    Article  MathSciNet  Google Scholar 

  23. Lü, X., Zhu, H.W., Meng, X.H., Yang, Z.C., Tian, B.: Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl. 336, 1305 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lü, X., Zhu, H.W., Yao, Z.Z., Meng, X.H., Zhang, C., Yang, Z.C., Tian, B.: Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications. Ann. Phys. 323, 1947 (2008)

    Article  MATH  Google Scholar 

  25. Lü, X., Li, L.L., Yao, Z.Z., Geng, T., Cai, K.J., Zhang, C., Tian, B.: Symbolic computation study of a generalized variable-coefficient two-dimensional Korteweg–de Vries model with various external-force terms from shallow water waves, plasma physics, and fluid dynamics. Z. Naturforsch. A, J. Phys. Sci. 64, 222 (2009)

    Google Scholar 

  26. Lü, X., Tian, B., Sun, K., Wang, P.: Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one Tau-function. J. Math. Phys. 51, 113506 (2010)

    Article  MathSciNet  Google Scholar 

  27. Lü, X., Geng, T., Zhang, C., Zhu, H.W., Meng, X.H., Tian, B.: Multi-soliton solutions and their interactions for the (2+1)-dimensional Sawada–Kotera model with truncated Painlevé expansion, Hirota bilinear method and symbolic computation. Int. J. Mod. Phys. B 23, 5003 (2009)

    Article  MATH  Google Scholar 

  28. Lü, X.: Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation. Chaos 23, 033137 (2013)

    Article  Google Scholar 

  29. Lü, X., Tian, B.: Vector bright soliton behaviors associated with negative coherent coupling. Phys. Rev. E 85, 026117 (2012)

    Article  Google Scholar 

  30. Lü, X., Tian, B.: Novel behavior and properties for the nonlinear pulse propagation in optical fibers. Europhys. Lett. 97, 10005 (2012)

    Article  Google Scholar 

  31. Lü, X., Tian, B.: Soliton solutions via auxiliary function method for a coherently-coupled model in the optical fiber communications. Nonlinear Anal.: Real World Appl. 14, 929 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rogers, C., Schief, W.K., Stallybrass, M.P.: Initial/boundary value problems and Darboux–Levi-type transformations associated with a 2+1-dimensional eigenfunction equation. Int. J. Non-Linear Mech. 30, 223 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Dubrovsky, V.G., Lisitsyn, Y.V.: The construction of exact solutions of two-dimensional integrable generalizations of Kaup–Kuperschmidt and Sawada–Kotera equations via ∂-dressing method. Phys. Lett. A 295, 198 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. Konopelchenko, B.G., Dubrovsky, V.G.: Some new integrable nonlinear evolution equations in 2+1 dimensions. Phys. Lett. A 102, 15 (1984)

    Article  MathSciNet  Google Scholar 

  35. Nucci, M.C.: Painleve property and pseudopotentials for nonlinear evolution equations. J. Phys. A 22, 2897 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zait, R.A.: Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations. Chaos Solitons Fractals 15, 673 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wazwaz, A.M.: The extended Tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 184, 1002 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Khater, A.H., Callebaut, D.K., Sayed, S.M.: Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces. J. Comput. Appl. Math. 189, 387 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

X.L. expresses his sincere thanks to Prof. X.B. Hu for his enthusiastic help and valuable discussions. This work is supported the National Natural Science Foundation of China under Grant No. 61308018, by China Postdoctoral Science Foundation under Grant No. 2012M520154, by the Fundamental Research Funds for the Central Universities (2013JBM088), and partially by the Project of State Key Laboratory of Rail Traffic Control and Safety (No. RCS2012ZT004), Beijing Jiao Tong University.

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiao Tong University, Beijing, 100044, P.R. China

    Xing Lü

  2. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao Tong University, Beijing, 100044, China

    Xing Lü

Authors
  1. Xing Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing Lü.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lü, X. New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada–Kotera model. Nonlinear Dyn 76, 161–168 (2014). https://doi.org/10.1007/s11071-013-1118-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-1118-y

Keywords

search

Navigation