Skip to main content
SpringerLink
Log in

Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations

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

Abstract

Interaction solutions are obtained from (2+1)-dimensional BLMP equation and (3+1)-dimensional nonlinear evolution equation by using a direct method. The interaction solutions of the two equations presented that a lump appear from a soliton wave and swallowed by it later, they are all the completely non-elastic interactions that rare to see. It shows that a lump appeared from the one side of the soliton wave (a kinky wave or another solitary wave), and separate slowly from this side of the soliton wave, after reaching the maximum separation at \(t=0,\) the lump gradually walks upon the other side of the soliton wave and finally swallowed by the other side. The dynamical character shows in this work enriches the variety of the dynamics of higher-dimensional nonlinear wave field.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. 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. JETP 77(2), 237–245 (1993)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lü, Z., Chen, Y.: Construction of rogue wave and lump solutions for nonlinear evolution equations. Eur. Phys. J. B 88, 187 (2015)

    Article  MathSciNet  Google Scholar 

  6. Ma, W.X., Qin, Z.Y., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  9. Ma, W.X., You, Y.: Solving the Korteweg-de Vries equation by its bilinear form: wronskian solutions. Trans. Am. Math. Soc. 357, 1753–1778 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ma, W.X., Li, C.X., He, J.: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. 70, 4245–4258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Nistazakis, H.E., Frantzeskakis, D.J., Malomed, B.A.: Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide. Phys. Rev. E 64, 026604 (2001)

    Article  Google Scholar 

  13. Wang, C.J., Dai, Z.D., Liu, C.F.: Interaction between kink solitary wave and Rogue wave for (2+1)-dimensional Burgers equation. Mediterr. J. Math. 13, 1087–1098 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Wazwaz, A.M.: (2+1)-dimensional Burgers equations BE(m+n+1), using the recursion operator. Appl. Math. Comput. 219, 9057–9068 (2013)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  18. Gilson, C.R., Nimmo, J.J.C., Willox, R.: A (2 + 1)-dimensional generalization of the AKNS shallow water wave equation. Phys. Lett. A 180, 337–345 (1993)

    Article  MathSciNet  Google Scholar 

  19. Luo, L.: New exact solutions and Bäcklund transformation for Boiti–Leon–Manna–Pempinelli equation. Phys. Lett. A 375, 1059–1063 (2011)

  20. Li, Y., Li, D.: New exact solutions for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Appl. Math. Sci. 6, 579–587 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Tang, Y., Zai, W.: New periodic-wave solutions for (2+1)- and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equations. Nonlinear Dyn. 81, 249–255 (2015)

    Article  MathSciNet  Google Scholar 

  22. Geng, X.G.: Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations. J. Phys. A Math. Gen. 36, 2289–2303 (2003)

    Article  MATH  Google Scholar 

  23. Geng, X., Ma, Y.: N-soliton solution and its Wronskian form of a (3+1)-dimensional nonlinear evolution equation. Phys. Lett. A 369, 285–289 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wazwaz, A.M.: A (3 + 1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions. Appl. Math. Comput. 215, 1548–1552 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Wazwaz, A.M.: New (3+1)-dimensional nonlinear evolution equation: multiple soliton solutions. Cent. Eur. J. Eng. 4, 352–356 (2014)

    Google Scholar 

  27. Wu, J.P.: A simple approach to derive a novel N-soliton solution for a (3+1)-dimensional nonlinear evolution equation. Commun. Theor. Phys. 53, 812–814 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu, J.P.: A new wronskian condition for a (3+1)-dimensional nonlinear evolution equation. Chin. Phys. Lett. 28, 50501–50503 (2011)

    Article  Google Scholar 

  29. Wu, J.P.: A generalized Hirota ansatz to obtain soliton-like solutions for a (3+1)-dimensional nonlinear evolution equation. Commun. Theor. Phys. 56, 297–300 (2011)

    Article  MATH  Google Scholar 

  30. Liu, N., Liu, Y.: New multi-soliton solutions of a (3+1)-dimensional nonlinear evolution equation. Comput. Math. Appl. 71, 1645–1654 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, Shanxi, China

    Yaning Tang, Siqiao Tao, Meiling Zhou & Qing Guan

Authors
  1. Yaning Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siqiao Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meiling Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qing Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaning Tang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, Y., Tao, S., Zhou, M. et al. Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations. Nonlinear Dyn 89, 429–442 (2017). https://doi.org/10.1007/s11071-017-3462-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3462-9

Keywords

Search

Navigation