Skip to main content
SpringerLink
Log in

Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

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

Abstract

We consider the simplified (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. We use the binary Bell polynomial theory to construct a bilinear form of the equation, and then construct a bilinear form of the special case of \(z = x\). In the reduced bilinear form, we constructed a more general lump solution that is positioned in any direction of the space to have more arbitrary autocephalous parameters. The lump solution can produce striped solitons, which provides a lumpoff solution. Combined with the strip solitons, we can know that when the double solitons cut the lump solution, we obtain a special rogue waves. It can be seen from our research results that the time and place of the rogue wave can be captured by tracking the moving path of the lump solution.

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. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)

    Article  MathSciNet  Google Scholar 

  2. Müller, P., Garrett, C., Osborne, A.: Rogue waves. Oceanography 18, 66–75 (2005)

    Article  Google Scholar 

  3. Yan, Z.: Rogue waves and their generating mechanisms in different physical contexts. Soc. Politics 18, 441–468 (2011)

    Article  Google Scholar 

  4. Shats, M., Punzmann, H., Xia, H.: Capillary rogue waves. Phys. Rev. Lett. 104, 104503 (2010)

    Article  Google Scholar 

  5. Ginzburg, N.S., Rozental, R.M., Sergeev, A.S., Fedotov, A.E., Zotova, I.V., Tarakanov, V.P.: Generation of rogue waves in gyrotrons operating in the regime of developed turbulence. Phys. Rev. Lett. 119(3), 034801 (2017)

    Article  Google Scholar 

  6. Moslem, W.M., Shukla, P.K., Eliasson, B.: Surface plasma rogue waves. EPL (Europhys. Lett.) 96, 25002 (2011)

    Article  Google Scholar 

  7. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80(3), 033610 (2009)

    Article  Google Scholar 

  8. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Vector rogue waves in binary mixtures of Bose–Einstein condensates. Eur. Phys. J. Spec. Top. 185, 169–180 (2010)

    Article  Google Scholar 

  9. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)

    Article  Google Scholar 

  10. Höhmann, R., Kuhl, U., Stöckmann, H.J., Kaplan, L., Heller, E.J.: Freak waves in the linear regime: a microwave study. Phys. Rev. Lett. 104(9), 093901 (2010)

    Article  Google Scholar 

  11. Montina, A., Bortolozzo, U., Residori, S., Arecchi, F.T.: Non-Gaussian statistics and extreme waves in a nonlinear optical cavity. Phys. Rev. Lett. 103(17), 173901 (2009)

    Article  Google Scholar 

  12. Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P.A.E.M., Kinoshita, T., Monbaliu, J., Mori, N., Osborne, A.R., Serio, M., Stansberg, C.T., Tamura, H., Trulsen, K.: Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Phys. Rev. Lett. 102(11), 114502 (2009)

    Article  Google Scholar 

  13. Wazwaz, A.M.: Partial Differential Equations: Methods and Applications. Balkema Publishers, Delft (2002)

    MATH  Google Scholar 

  14. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wazwaz, A.M., Xu, G.Q.: Negative-order modified KdV equations: multiple soliton and multiple singular soliton solutions. Math. Meth. Appl. Sci. 39(4), 661–667 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ganshin, A.V., Efimov, V.B., Kolmakov, G.V., Mezhov-Deglin, L.P., McClintock, P.V.E.: Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. Phys. Rev. Lett. 101(6), 065303 (2008)

    Article  Google Scholar 

  17. Ma, H.C., Deng, A.P.: Lump solution of (2+1)-dimensional Boussinesq equation. Commun. Theor. Phys. 65, 546–552 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85(2), 1217–1222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, X., Chen, Y., Tang, X. Y.: Rogue wave and a pair of resonance stripe solitons to a reduced generalized (3+1)-dimensional KP equation. arXiv:1610.09507 [nlin.SI] (2016)

  24. Liu, Y.K., Li, B.: Dynamics of rogue waves on multisoliton background in the Benjamin Ono equation. Pramana J. Phys. 88, 57 (2017)

    Article  Google Scholar 

  25. Chen, M., Li, B.: Hybrid soliton solutions in the (2+1)-dimensional nonlinear Schrödinger equation. Mod. Phys. Lett. B 32, 1750298 (2017)

    Article  Google Scholar 

  26. Li, X., Wang, Y., Chen, M., Li, B.: Lump solutions and resonance stripe solitons to the (2+1)-dimensional Sawada-Kotera equation. Adv. Math. Phys. 2017, 1743789 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Peng, W.Q., Tian, S.F., Zhang, T.T.: Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation. EPL (Europhys. Lett.) 123(5), 50005 (2018)

    Article  MathSciNet  Google Scholar 

  28. Qin, C.Y., Tian, S.F., Zou, L., Ma, W.X.: Solitary wave and quasi-periodic wave solutions to a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. Adv. Appl. Math. Mech. 10(4), 948–977 (2018)

    Article  MathSciNet  Google Scholar 

  29. Wang, X.B., Tian, S.F., Feng, L.L., Zhang, T.T.: On quasi-periodic waves and rogue waves to the (4+1)-dimensional nonlinear Fokas equation. J. Math. Phys. 59(7), 073505 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yan, X.W., Tian, S.F., Dong, M.J., Zhou, L., Zhang, T.T.: Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation. Comput. Math. Appl. 76(1), 179–186 (2018)

    Article  MathSciNet  Google Scholar 

  31. Qin, C.Y., Tian, S.F., Wang, X.B., Zhang, T.T., Li, J.: Rogue waves, bright-dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Comput. Math. Appl. 75(12), 4221–4231 (2018)

    Article  MathSciNet  Google Scholar 

  32. Dong, M.J., Tian, S.F., Yan, X.W., Zou, L.: Solitary waves, homoclinic breather waves and rogue waves of the (3+1)-dimensional Hirota bilinear equation. Comput. Math. Appl. 75(3), 957–964 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, X.B., Tian, S.F., Zhang, T.T.: Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation. Proc. Am. Math. Soc. 146(8), 3353–3365 (2018)

    Article  MATH  Google Scholar 

  34. Wang, X.B., Zhang, T.T., Dong, M.J.: Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation. Appl. Math. Lett. 86, 298–304 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Feng, L.L., Zhang, T.T.: Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation. Appl. Math. Lett. 78, 133–140 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yang, J.Y., Ma, W.X.: Abundant interaction solutions of the KP equation. Nonlinear Dyn. 89, 1539–1544 (2017)

    Article  MathSciNet  Google Scholar 

  37. Tang, X., Lou, S., Zhang, Y.: Localized excitations in (2+1)-dimensional systems. Phys. Rev. E 66(4), 046601 (2002)

    Article  MathSciNet  Google Scholar 

  38. Yang, J.Y., Ma, W.X.: Abundant interaction solutions of the KP equation. Nonlinear Dyn. 89(2), 1539–1544 (2017)

    Article  MathSciNet  Google Scholar 

  39. Frenkel, E., Losev, A., Nekrasov, N.: Instantons beyond topological theory. I. J. Inst. Math. Jussieu 10, 463–465 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Frenkel, E., Losev, A., Nekrasov, N.: Notes on instantons in topological field theory and beyond. Nucl. Phys. B Proc. Suppl. 171, 215–230 (2007)

    Article  MathSciNet  Google Scholar 

  41. Sun, Y., Tian, B., Xie, X.Y., Chai, J., Yin, H.M.: Rogue waves and lump solitons for a-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics. Waves Random Complex Media 28, 544–552 (2018)

    Article  MathSciNet  Google Scholar 

  42. Huang, Z.R., Tian, B., Zhen, H.L.: Bäcklund transformations and soliton solutions for a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics. Nonlinear Dyn. 80, 1–7 (2015)

    Article  MATH  Google Scholar 

  43. Wazwaz, A.M.: (2+1)-dimensional Korteweg-de Vries(N) equations derived by using the Korteweg-de Vries recursion operator. Phys Scr. 86, 035007 (2012)

    Article  MATH  Google Scholar 

  44. Abudiab, M., Khalique, C.M.: Exact solutions and conservation laws of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. Adv. Differ. Equ. 2013, 1–7 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gao, X.Y.: Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid mechanics. Ocean Eng. 96, 245–247 (2015)

    Article  Google Scholar 

  46. Gilson, C., Lambert, F., Nimmo, J., Willox, R.: On the Combinatorics of the Hirota D-Operators. Proc. R. Soc. Lond. A 452, 223–234 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lambert, F., Springael, J.: Soliton equations and simple combinatorics. Acta Appl. Math. 102, 147–178 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Fan, E.: The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials. Phys. Lett. A 375, 493–497 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  50. Peng, W.Q., Tian, S.F., Zhang, T.T.: Analysis on lump, lumpoff and rogue waves with predictability to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation. Phys. Lett. A 241, 1–8 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Wazwaz, A.M.: Multiple-soliton solutions for the fifth-order Caudrey–Dodd–Gibbon equation. Appl. Math. Comput. 197, 719–724 (2008)

    MathSciNet  MATH  Google Scholar 

  52. Wazwaz, A.M.: Multiple soliton solutions for (2+1)-dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations. Math. Meth. Appl. Sci. 34, 1580–1586 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  53. 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 

  54. Wang, D.S., Wei, X.Q.: Integrability and exact solutions of a two-component Korteweg-de Vries system. Appl. Math. Lett. 51, 60 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  55. Dai, C.Q., Huang, W.H.: Multi-rogue wave and multi-breather solutions in PT-symmetric coupled waveguides. Appl. Math. Lett. 32, 35 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yu, F.J.: Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz–Musslimani equation with PT-symmetric potential. Chaos 27, 023108 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  57. Qin, C.Y., Tian, S.F., Zou, L., Zhang, T.T.: Lie symmetry analysis, conservation laws and exact solutions of fourth-order time fractional Burgers equation. J. Appl. Anal. Comput. 8(6), 1727–1746 (2018)

    MathSciNet  Google Scholar 

  58. Tian, S.F.: Infinite propagation speed of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett. 89, 1–7 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tian, S.F.: Asymptotic behavior of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett. 83, 65–72 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  60. Tian, S.F.: Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Commun. Pure Appl. Anal. 17(3), 923–957 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  61. Tian, S.F., Zhang, T.T.: Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proc. Am. Math. Soc. 146(4), 1713–1729 (2018)

    Article  MATH  Google Scholar 

  62. Tian, S.F.: Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method. J. Phys. A Math. Theor. 50(39), 395204 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  63. Tian, S.F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method. J. Differ. Equ. 262(1), 506–558 (2017)

    Article  MATH  Google Scholar 

  64. Tian, S.F.: The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method. Proc. R. Soc. Lond. A 472(2195), 20160588 (2016)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the editor and reviewers for their valuable comments. This work was supported by the Jiangsu Province Natural Science Foundation of China under Grant No. BK20181351, the Postgraduate Research & Practice Program of Education and Teaching Reform of CUMT under Grant No. YJSJG_2018_036, the “Qinglan Engineering project” of Jiangsu Universities, the No. [2016] 22 supported by Ministry of Industry and Information Technology of China, the National Natural Science Foundation of China under Grant Nos. 11301527 and 51522902, the Fundamental Research Funds for the Central Universities under Grant Nos. DUT17ZD233 and 2017XKQY101, and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M570498 and 2017T100413.

Author information

Authors and Affiliations

  1. School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Jin-Jin Mao, Shou-Fu Tian, Tian-Tian Zhang & Xing-Jie Yan

  2. School of Naval Architecture, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Li Zou

  3. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, People’s Republic of China

    Li Zou

Authors
  1. Jin-Jin Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shou-Fu Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tian-Tian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xing-Jie Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Zou.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mao, JJ., Tian, SF., Zou, L. et al. Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. Nonlinear Dyn 95, 3005–3017 (2019). https://doi.org/10.1007/s11071-018-04736-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-04736-2

Keywords

Search

Navigation