Nonlinear Dynamics

, Volume 95, Issue 1, pp 29–42 | Cite as

Interaction phenomena between a lump and other multi-solitons for the \({\mathbf {(2+1)}}\)-dimensional BLMP and Ito equations

  • Chunhua He
  • Yaning TangEmail author
  • Wenxiu Ma
  • Jinli Ma
Original Paper


In this paper, “new” interaction solutions between a lump solution and other multi-soliton (kinky or stripe) solutions are studied through developing a “new” direct method based on the Hirota bilinear form for the \((2+1)\)-dimensional BLMP equation and the \((2+1)\)-dimensional Ito equation. Interaction solutions degenerate into lump (or kinky/stripe) solutions while the involved exponential function (or quadratic function) disappears. The interaction phenomena in the presented solutions show that a lump can be drowned or swallowed by other multi-solitary waves (kinky or stripe waves), and such interactions are very rare non-elastic collisions. What is more, we find that the positions of the interaction between a lump and three or four kinky waves are different while we choose different parameters, and the collisions may be at the bottom, middle, top or other positions. The dynamic characteristics of the constructed interaction solutions are illustrated by sequences of interesting figures plotted with the help of Maple.


Lump solution Soliton solution Interaction solution The \((2+1)\)-dimensional BLMP equation The \((2+1)\)-dimensional Ito equation 



We are very grateful to the editor and reviewers for their constructive comments and suggestions. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. This research is supported by the Natural Science Basic Research Program of Shaanxi (Grant No. 2017JM1024) and sponsored by the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Grant No. ZZ2018174).


  1. 1.
    Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction. Phys. Lett. A 63(3), 205–206 (1977)CrossRefGoogle Scholar
  2. 2.
    Nakamura, A.: Explode-decay mode lump solitons of a two-dimensional nonlinear Schrödinger equation. Phys. Lett. A 88(2), 55–56 (1982)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Obukhov, Y.N., Vlachynsky, E.J., Esser, W., Tresguerres, R., Hehl, F.W.: An exact solution of the metric-affine gauge theory with dilation, shear, and spin charges. Phys. Lett. A 220(1–3), 1–9 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Zhu, X.M., Zhang, D.J., Chen, D.Y., Zhu, X.M., Chen, D.Y.: Lump solutions of Kadomtsev–Petviashvili I equation in non-uniform media. Theor. Phys. 55(1), 13–19 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Lu, Z., Chen, Y.: Construction of rogue wave and lump solutions for nonlinear evolution equations. Eur. Phys. J. B 88(7), 1–5 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Imai, K., Nozaki, K.: Lump solutions of the Ishimori-II equation. Prog. Theor. Phys. 96(3), 521–526 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379(36), 1975–1978 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Wu, X.H., He, J.H.: Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method. Comput. Math. Appl. 54(7), 966–986 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Zhang, L., Huo, X.: On the exp-function method for constructing travelling wave solutions of nonlinear equations. Nonlinear Mod. Math. Phys. 1212, 280–285 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ma, W.X., Huang, T.W., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82(6), 5468–5478 (2010)CrossRefGoogle Scholar
  11. 11.
    Tang, Y., Tao, S., 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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Tang, Y., Tao, S., Zhou, M., Guan, Q.: Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations. Nonlinear Dyn. 89(2), 1–14 (2017)MathSciNetGoogle Scholar
  13. 13.
    Huang, L., Chen, Y.: Lump solutions and interaction phenomenon for \((2+1)\)-dimensional Sawada–Kotera equation. Theor. Phys. 67(5), 473–478 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Tan, W., Dai, Z., Xie, J., Hu, L.: Emergence and interaction of the lump-type solution with the \((3+1)\)-d Jimbo–Miwa equation. Zeitschrift Fr Naturforschung A 73(1), 43–49 (2017)Google Scholar
  15. 15.
    Wang, Y., Chen, M.D., Li, X., Li, B.: Some interaction solutions of a reduced generalised \((3+1)\)-dimensional shallow water wave equation for lump solutions and a pair of resonance solitons. Zeitschrift Fr Naturforschung A 72(5), 419–424 (2017)Google Scholar
  16. 16.
    Kofane, T.C., Fokou, M., Mohamadou, A., Yomba, E.: Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation. Eur. Phys. J. Plus 132(11), 465 (2017)CrossRefGoogle Scholar
  17. 17.
    Ahmed, I.: Interaction solutions for lump-line solitons and lump-kink waves of the dimensionally reduced generalised KP equation. Zeitschrift Fr Naturforschung A 72(10), 955–961 (2017)Google Scholar
  18. 18.
    Nistazakis, H.E., Frantzeskakis, D.J., Malomed, B.A.: Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 64(2) (2001).
  19. 19.
    Lu, Z., Tian, E.M., Grimshaw, R.: Interaction of two lump solitons described by the Kadomtsev–Petviashvili I equation. Wave Motion 40(2), 123–135 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fokas, A.S., Pelinovsky, D.E., Sulem, C.: Interaction of lumps with a line soliton for the dsii equation. Phys. D Nonlinear Phenom. 152–153(3), 189–198 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Wang, C., Dai, Z., Liu, C.: Interaction between kink solitary wave and rogue wave for \((2+1)\)-dimensional Burgers equation. Mediterr. J. Math. 13(3), 1087–1098 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Tan, W., Dai, Z.: Dynamics of kinky wave for \((3+1)\)-dimensional potential Yu–Toda–Sasa–Fukuyama equation. Nonlinear Dyn. 85(2), 817–823 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zheltukhin, A.N., Flegel, A.V., Frolov, M.V., Manakov, N.L., Starace, A.F.: Rescattering effects in laser-assisted electron–atom bremsstrahlung. J. Phys. B At. Mol. Opt. Phys. 48(7), 75,202–75,216(15) (2015)CrossRefGoogle Scholar
  24. 24.
    Conn, R.W., Kesner, J.: Plasma modeling and first wall interaction phenomena in tokamaks. J. Nucl. Mater. 63(1), 1–14 (1976)CrossRefGoogle Scholar
  25. 25.
    Adelberger, E.G.: Weak interaction experiments at low energies results from atomic and nuclear physics. AIP Conf. Proc. 81, 259–279 (1982)CrossRefGoogle Scholar
  26. 26.
    Fabris, G., Hantman, R.G.: Interaction of fluid dynamics phenomena and generator efficiency in two-phase liquid-metal gas magnetohydrodynamic power generators. Energy Convers. Manag. 21(1), 49–60 (1981)CrossRefGoogle Scholar
  27. 27.
    Garc-Alvarado, M.G., Flores-Espinoza, R., OmelYanov, G.A.: Interaction of shock waves in gas dynamics: uniform in time asymptotics. Int. J. Math. Math. Sci. 2005(19), 3111–3126 (2014)Google Scholar
  28. 28.
    Slowman, A.B., Evans, M.R., Blythe, R.A.: Exact solution of two interacting run-and-tumble random walkers with finite tumble duration. J. Phys. A Math. Theor. 50(37) (2017).
  29. 29.
    Kaatze, U.: Electromagnetic Wave Interactions with Water and Aqueous Solutions. Springer, Berlin (2005)CrossRefGoogle Scholar
  30. 30.
    Song, L., Pu, L., Dai, Z.: Spatio-temporal deformation of kink-breather to the \((2+1)\)-dimensional potential Boiti–Leon–Manna–Pempinelli equation. Theor. Phys. 67(5), 493–497 (2017)Google Scholar
  31. 31.
    Yang, J.Y., Ma, W.X., Qin, Z.Y.: Lump and lump-soliton solutions to the \((2+1)\)-dimensional Ito equation. Anal. Math. Phys. 8, 427–436 (2018)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Hirota, R.: Direct Methods in Soliton Theory. Springer, Berlin (1980)CrossRefGoogle Scholar
  33. 33.
    Gilson, C.R., Nimmo, J., Willox, R.: A \((2+1)\)-dimensional generalization of the AKNS shallow water wave equation. Phys. Lett. A 180(4–5), 337–345 (1993)MathSciNetGoogle Scholar
  34. 34.
    Li, Y., Li, D.: New exact solutions for the \((2+1)\)-dimensional Boiti–Leon–Manna–Pempinelli equation. Appl. Math. Sci. 30(1), 579–587 (2012)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Tang, Y., Zai, W.: New periodic-wave solutions for \((2+1)\)-and \((3+1)\)-dimensional Boiti–Leon–Manna–Pempinelli equations. Nonlinear Dyn. 81(1–2), 249–255 (2015)MathSciNetGoogle Scholar
  36. 36.
    Ito, M.: An extension of nonlinear evolution equations of the K-dv (mK-dv) type to higher orders. J. Phys. Soc. Jpn. 49(2), 771–778 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Wazwaz, A.M.: Multiple-soliton solutions for the generalized \((1+1)\)-dimensional and the generalized \((2+1)\)-dimensional Ito equations. Appl. Math. Comput. 202(2), 840–849 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Chunhua He
    • 1
  • Yaning Tang
    • 1
    Email author
  • Wenxiu Ma
    • 2
    • 3
    • 4
  • Jinli Ma
    • 1
  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  3. 3.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  4. 4.Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical ModellingNorth-West UniversityMmabathoSouth Africa

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