Advertisement

Nonlinear Dynamics

, Volume 93, Issue 4, pp 1841–1851 | Cite as

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

  • Wei-Qi Peng
  • Shou-Fu Tian
  • Li Zou
  • Tian-Tian Zhang
Original Paper
  • 154 Downloads

Abstract

In this paper, we consider a (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.

Keywords

The (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation Hirota bilinear form Interaction waves Lump waves Soliton waves 

Notes

Acknowledgements

The authors would like to thank the editor and the referees for their valuable comments and suggestions. This work was supported by the Research and Practice of Educational Reform for Graduate students in China University of Mining and Technology under Grant No. YJSJG_2017_049, the No. [2016] 22 supported by Ministry of Industry and Information Technology of China, the Qinglan Engineering project of Jiangsu Universities, the National Natural Science Foundation of China under Grant Nos. 11301527 and 51522902, the Fundamental Research Funds for the Central Universities under Grant No. DUT17ZD233, and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M570498 and 2017T100413.

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the authors.

References

  1. 1.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  2. 2.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
  3. 3.
    Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  4. 4.
    Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ma, W.X., Huang, T.W., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82, 065003 (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Wazwaz, A.M.: Partial Differential Equations: Methods and Applications. Balkema Publishers, Amsterdam (2002)MATHGoogle Scholar
  7. 7.
    Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Wazwaz, A.M., Xu, G.Q.: Negative-ordermodified KdV equations: multiple soliton andmultiple singular soliton solutions. Math. Methods Appl. Sci. 39(4), 661–667 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Estevez, P.G., Diaz, E., Dominguez-Adame, F., et al.: Lump solitons in a higher-order nonlinear equation in (\(2+1\))-dimensions. Phys. Rev. E 93, 062219 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Singh, N., Stepanyants, Y.: Obliquely propagating skew KP lumps. Wave Motion 64, 92–102 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A. 379, 1975–1978 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ma, W.X., Qin, Z., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zhao, H.Q., Ma, W.X.: Mixed lumpCkink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (\(3+1\))-dimensional Kadomtsev–Petviashvili equation. Appl. Math. Lett 72, 58–64 (2017)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Yang, J.Y., Ma, W.X., Qin, Z.Y.: Lump and lump-soliton solutions to the (\(2+1\))-dimensional Ito equation. Anal. Math. Phys.  https://doi.org/10.1007/s13324-017-0181-9
  18. 18.
    Huang, L.L., Chen, Y.: Lump solutions and interaction phenomenon for (\(2+1\))-dimensional Sawada–Kotera equation. Commun. Theor. Phys. 67(5), 473 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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)
  20. 20.
    Chen, M.D., Li, X., Wang, Y., Li, B.: A pair of resonance stripe solitons and lump solutions to a reduced (\(3+1\))-dimensional nonlinear evolution equation. Commun. Theor. Phys. 67(6), 595 (2017)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jia, M., Lou, S.Y.: A novel type of rogue waves with predictability in nonlinear physics, nlin. PS (2017)Google Scholar
  22. 22.
    Konopelchenko, B.G., Dubrovsky, V.G.: Some new integrable nonlinear evolution equations in \(2+1\) dimensions. Phys. Lett. A 102, 15 (1984)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sawada, K., Kotera, J.: A method for finding N-soliton solutions of the K.d.V. equation and K.d.V.-like equation. Prog. Theor. Phys. 51, 1355 (1974)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Cheng, Y., Li, Y.S.: Constraints of the \(2+1\) dimensional integrable soliton systems. J. Phys. A 25, 419 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Cao, C.W., Wu, Y.T., Geng, X.G.: On quasi-periodic solutions of the \(2+1\) dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation. Phys. Lett. A 256, 59–65 (1999)CrossRefGoogle Scholar
  26. 26.
    Wazwaz, A.M.: Multiple-soliton solutions for the fifth-order Caudrey–Dodd–Gibbon equation. Appl. Math. Comput. 197, 719–724 (2008)MathSciNetMATHGoogle Scholar
  27. 27.
    Wazwaz, A.M.: Multiple soliton solutions for (\(2+1\))-dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations. Math. Methods Appl. Sci. 34, 1580–1586 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wazwaz, A.M., El-Tantawy, S.A.: Solving the (\(3+1\))-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear. Dyn. 88, 3017–3021 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tian, S.F.: The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method. Proc. R. Soc. Lond. A 472, 20160588 (2016). (22pp)CrossRefMATHGoogle Scholar
  30. 30.
    Tian, S.F.: Initial-boundary value problems for the coupled modified Korteweg–de Vries equation on the interval. Commun. Pure Appl. Anal. 173, 923–957 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    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)CrossRefMATHGoogle Scholar
  32. 32.
    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, 395204 (2017)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tu, J.M., Tian, S.F., Xu, M.J., Zhang, T.T.: On Lie symmetries, optimal systems and explicit solutions to the Kudryashov–Sinelshchikov equation. Appl. Math. Comput. 275, 345–352 (2016)MathSciNetGoogle Scholar
  34. 34.
    Tian, S.F., Zhang, Y.F., Feng, B.L., Zhang, H.Q.: On the Lie algebras, generalized symmetries and Darboux transformations of the fifth-order evolution equations in shallow water. Chin. Ann. Math. B 36(4), 543–560 (2015)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Tian, S.F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equations on the interval via the Fokas method. J. Differ. Equ. 262, 506–558 (2017)CrossRefMATHGoogle Scholar
  36. 36.
    Ma, W.X., Qin, Z.Y., La, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–31 (2016)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    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)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Ma, W.X., Li, C.X., He, J.: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. 70, 4245–4258 (2009)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wang, D.S., Wei, X.Q.: Integrability and exact solutions of a two-component Korteweg–de Vries system. Appl. Math. Lett. 51, 60 (2016)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Dai, C.Q., Huang, W.H.: Multi-rogue wave and multi-breather solutions in PT-symmetric coupled waveguides. Appl. Math. Lett. 32, 35 (2014)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Yu, F.J.: Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz–Musslimani equation with PT-symmetric potential. Chaos 27, 023108 (2017)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Tian, S.F.: Asymptotic behavior of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett. 83, 65–72 (2018)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Feng, L.L., Tian, S.F., Zhang, T.T., Zhou, J.: Nonlocal symmetries, consistent Riccati expansion, and analytical solutions of the variant Boussinesq system. Z. Naturforsch. A 72(7), 655–663 (2017)CrossRefGoogle Scholar
  44. 44.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Lie symmetry analysis, analytical solutions, and conservation laws of the generalised Whitham–Broer–Kaup-like equations. Z. Naturforsch. A 72(3), 269–279 (2017)CrossRefGoogle Scholar
  45. 45.
    Feng, L.L., Tian, S.F., Zhang, T.T.: Nonlocal symmetries and consistent Riccati expansions of the (\(2+ 1\))-dimensional dispersive long wave equation. Z. Naturforsch. A 72(5), 425–431 (2017)CrossRefGoogle Scholar
  46. 46.
    Tian, S.F., Zhang, H.Q.: Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations. J. Math. Anal. Appl. 371, 585–608 (2010)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Tian, S.F., Zhang, H.Q.: On the integrability of a generalized variable-coefficient forced Korteweg–de Vries equation in fluids. Stud. Appl. Math. 132, 212 (2014)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Tian, S.F., Zhang, H.Q.: On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. J. Phys. A Math. Theor. 45, 055203 (2012)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Tu, J.M., Tian, S.F., Xu, M.J., Song, X.Q., Zhang, T.T.: Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (\(3+1\))-dimensional nonlinear wave in liquid with gas bubbles. Nonlinear Dyn. 83, 1199–1215 (2016)CrossRefMATHGoogle Scholar
  50. 50.
    Tu, J.M., Tian, S.F., Xu, M.J., Zhang, T.T.: Quasi-periodic waves and solitary waves to a generalized KdV-Caudrey–Dodd–Gibbon equation from fluid dynamics. Taiwanese J. Math. 20, 823–848 (2016)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Tian, S.F., Zhang, H.Q.: Riemann theta functions periodic wave solutions and rational characteristics for the (\(1+1\))-dimensional and (\(2+1\))-dimensional Ito equation. Chaos Solitons Fractals 47, 27 (2013)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Xu, M.J., Tian, S.F., Tu, J.M., Zhang, T.T.: Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (\(2+1\))-dimensional Boussinesq equation. Nonlinear Anal. Real World Appl. 31, 388–408 (2016)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Xu, M.J., Tian, S.F., Tu, J.M., Ma, P.L., Zhang, T.T.: On quasiperiodic wave solutions and integrability to a generalized (\(2+1\))-dimensional Korteweg–de Vries equation. Nonlinear Dyn. 82, 2031–2049 (2015)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Tu, J.M., Tian, S.F., Xu, M.J., Ma, P.L., Zhang, T.T.: On periodic wave solutions with asymptotic behaviors to a (\(3+1\))-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics. Comput. Math. Appl. 72, 2486–2504 (2016)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Wang, X.B., Tian, S.F., Xu, M.J., Zhang, T.T.: On integrability and quasi-periodic wave solutions to a (\(3+1\))-dimensional generalized KdV-like model equation. Appl. Math. Comput. 283, 216–233 (2016)MathSciNetGoogle Scholar
  56. 56.
    Wang, X.B., Tian, S.F., Feng, L.L., Yan, H., Zhang, T.T.: Quasiperiodic waves, solitary waves and asymptotic properties for a generalized (\(3+1\))-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation. Nonlinear Dyn. 88, 2265–2279 (2017)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Tian, S.F., Zhang, H.Q.: A kind of explicit Riemann theta functions periodic wave solutions for discrete soliton equations. Commun. Nonlinear Sci. Numer. Simul. 16, 173–186 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Institute of Mathematical PhysicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.School of Naval Architecture, State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianPeople’s Republic of China
  3. 3.Collaborative Innovation Center for Advanced Ship and Deep-Sea ExplorationShanghaiPeople’s Republic of China

Personalised recommendations