Lump solutions and bilinear Bäcklund transformation for the \((4+1)\)-dimensional Fokas equation

Abstract

The \((4+1)\)-dimensional Fokas (4DFK) equation is explicitly written out by means of the linearized operator of the 4DFK equation after introducing an additional auxiliary variable. Based on the bilinear form, bilinear Bäcklund transformations consisting of three bilinear equations are provided. A new class of lump solutions is given and analyzed using the extremum theory of multivariate function, of which the peculiar dynamic property is demonstrated graphically.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

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

    Google Scholar 

  2. 2.

    Hietarinta, J.: Hirota’s bilinear method and soliton solutions. Phys. AUC 15(Part 1), 31–37 (2005)

    Google Scholar 

  3. 3.

    Hu, X.B.: The Bäcklund transformations and nonlinear superposition formula of a modified Korteweg–De Vries-type bilinear equation. J. Nonlinear Math. Phys. 35, 4739–4745 (1994)

    MATH  Article  Google Scholar 

  4. 4.

    Gorshkov, K.A., Pelinovsky, D.E., Stepanyants, Y.A.: Normal and anomalous scattering, formation and decay of bound states of two-dimensional solitons described by the Kadomtsev-Petviashvili equation. J. Exp. Theor. Phys. 77(2), 237–245 (1993)

    Google Scholar 

  5. 5.

    Manakov, S.V., Zakhorov, V.E., Bordag, L.A., et al.: Two-dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction. Phys. Lett. A. 63, 205–206 (1977)

    Article  Google Scholar 

  6. 6.

    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A. 379, 1975–1978 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Lu, Z., Tian, E.M., Grimshaw, R.: Interaction of two lump solitons described by the Kadomtsev–Petviashvili I equation. Wave Motion 40, 123–135 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Lü, Z.S., Chen, Y.N.: Construction of rogue wave and lump solutions for nonlinear evolution equations. Eur. Phys. J. B. 88(7), 1–5 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Ma, W.X.: A search for lump solutions to a combined fourth-order nonlinear PDE in \((2+1)\)-dimensions. J. Appl. Anal. Comput. 9(4), 1319–1332 (2019)

    MathSciNet  Google Scholar 

  11. 11.

    Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B. 30, 1640028 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Ma, W.X.: Lump-type solutions to the \((3+1)\)-dimensional Jimbo–Miwa equation. Int. J. Nonlinear Sci. Num. 17, 355–359 (2016)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Ma, W.X., Yong, X.L., Zhang, H.Q.: Diversity of interaction solutions to the \((2+1)\)-dimensional Ito equation. Comput. Math. Appl. 75(10), 289–295 (2018)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Fokas, A.S., Pelinovsky, D.E., Sulem, C.: Interaction of lumps with a line soliton for the DSII equation. Phys. D. 152–153, 189–198 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Tang, Y.N., Tao, S.Q., Guan, Q.: Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput. Math. Appl. 72(9), 2334–2342 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Tan, W., Dai, Z.D.: Dynamics of kinky wave for \((3+1)\)-dimensional potential Yu–Toda–Sasa–Fukuyama equation. Nonlinear Dyn. 85(2), 817–823 (2016)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Fokas, A.S.: Integrable nonlinear evolution PDEs in \(4+2\) and \(3+1\) dimensions. Phys. Rev. Lett. 96, 190201 (2006)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Yang, Z.Z., Yan, Z.Y.: Symmetry groups and exact solutions of new \((4+1)\)-dimensional Fokas equation. Commun. Theor. Phys. 51(5), 876–880 (2009)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Lee, J., Sakthivel, R., Wazzan, L.: Exact traveling wave solutions of a higher-dimensional nonlinear evolution equation. Mod. Phys. Lett. B. 24, 1011–1021 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    He, Y.H.: Exact solutions for \((4+1)\)-dimensional nonlinear Fokas equation using extended F-expansion method and its variant. Math. Probl. Eng. 2014, 1–11 (2014)

    MathSciNet  Google Scholar 

  21. 21.

    Zhang, S., Tian, C., Qian, W.Y.: Bilinearization and new multisoliton solutions for the \((4+1)\)-dimensional Fokas equation. Pramana J. Phys. 86, 1259–1267 (2016)

    Google Scholar 

  22. 22.

    Cheng, L., Zhang, Y.: Lump-type solutions for the \((4+1)\)-dimensional Fokas equation via symbolic computations. Mod. Phys. Lett. B. 31, 1750224 (2017)

    MathSciNet  Google Scholar 

  23. 23.

    Wang, C.J.: Spatiotemporal deformation of lump solution to \((2+1)\)-dimensional KdV equation. Nonlinear Dyn. 84, 697–702 (2016)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers and editors for their suggestions and helpful comments on improving the paper.

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Ruoxia Yao or Yali Shen.

Additional information

Publisher's Note

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

Supported by the National Natural Science Foundation of China (11471004, 11501498), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0868) and the Project of the Youth Fund of Shanxi (201901D211461)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yao, R., Shen, Y. & Li, Z. Lump solutions and bilinear Bäcklund transformation for the \((4+1)\)-dimensional Fokas equation. Math Sci (2020). https://doi.org/10.1007/s40096-020-00341-w

Download citation

Keywords

  • \((4+1)\)-Dimensional Fokas equation
  • Bilinear form
  • Bilinear Bäcklund transformation
  • Lump solutions