Nonlinear Dynamics

, Volume 97, Issue 4, pp 2023–2040 | Cite as

Breather and hybrid solutions for a generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation for the water waves

  • Cui-Cui Ding
  • Yi-Tian GaoEmail author
  • Gao-Fu Deng
Original paper


Water waves are one of the most common phenomena in nature, the study of which helps in designing the related industries. In this paper, a generalized (\(3+1\))-dimensional B-type Kadomtsev–Petviashvili equation for the water waves is investigated. Gramian solutions are constructed via the Kadomtsev–Petviashvili hierarchy reduction. Based on the Gramian solutions, we construct the breathers. We graphically analyze the breather solutions and find that the breathers can be reduced to the homoclinic orbits. For the higher-order breather solutions, we obtain the mixed solutions consisting of the breathers and homoclinic orbits. According to the long-wave limit method, rational solutions are constructed. We look at two types of the rational solutions, i.e., the lump and line rogue wave solutions, and give the condition for the lumps being reduced to the line rogue waves. Taking another set of the parameters for the Gramian solutions, we also derive the kinky breather solutions which can be reduced to the kink solitons. For the higher-order kinky breather solutions, we obtain the mixed solutions consisting of the breathers and kink solitons. Combining the breather and rational solutions, we construct two kinds of the hybrid solutions composed of the breathers, lumps, line rogue waves and kink solitons. Characteristics of those hybrid solutions are graphically analyzed and the conditions for the generation of those hybrid solutions are given.


Water waves Generalized (\(3+1\))-dimensional B-type Kadomtsev–Petviashvili hierarchy reduction Breather solutions Hybrid solutions 



We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023 and by the Fundamental Research Funds for the Central Universities under Grant No. 50100002016105010.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ablowitz, M.J., Segur, H.: On the evolution of packets of water waves. J. Fluid Mech. 92, 691–715 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Zhao, X.H., Tian, B., Guo, Y.J., Li, H.M.: Solitons interaction and integrability for a (\(2+1\))-dimensional variable-coefficient Broer-Kaup system in water waves. Mod. Phys. Lett. B 32, 1750268 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yin, H.M., Tian, B., Chai, J., Wu, X.Y.: Stochastic soliton solutions for the (\(2+1\))-dimensional stochastic Broer-Kaup equations in a fluid or plasma. Appl. Math. Lett. 82, 126 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dolatshah, A., Nelli, F., Bennetts, L.G.: Hydroelastic interactions between water waves and floating freshwater ice. Phys. Fluids 30, 091702 (2018)CrossRefGoogle Scholar
  6. 6.
    Lan, Z.Z., Hu, W.Q., Guo, B.L.: General propagation lattice Boltzmann model for a variable-coefficient compound KdV-Burgers equation. Appl. Math. Model. 73, 695–714 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Xie, X.Y., Meng, G.Q.: Dark solitons for a variable-coefficient AB system in the geophysical fluids or nonlinear optics. Eur. Phys. J. Plus 134, 359 (2019)CrossRefGoogle Scholar
  8. 8.
    Zhao, X.H., Tian, B., Xie, X.Y., Wu, X.Y., Sun, Y., Guo, Y.J.: Solitons, Backlund transformation and Lax pair for a (\(2+1\))-dimensional Davey-Stewartson system on surface waves of finite depth. Wave. Random Complex 28, 356 (2018)CrossRefGoogle Scholar
  9. 9.
    Sun, Y., Tian, B., Liu, L., Wu, X.Y.: Rogue waves for a generalized nonlinear Schrödinger equation with distributed coefficients in a monomode optical fiber. Chaos, Solitons Fractals 107, 266 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lan, Z.Z., Gao, B.: Lax pair, infinitely many conservation laws and solitons for a (\(2 +1\))-dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients. Appl. Math. Lett. 79, 6–12 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lan, Z.Z.: Multi-soliton solutions for a (\(2+ 1\))-dimensional variable-coefficient nonlinear Schrödinger equation. Appl. Math. Lett. 86, 243–248 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hammack, J., Scheffner, N., Segur, H.: Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567–589 (1989)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wang, M., Tian, B., Sun, Y., Yin, H.M., Zhang, Z.: Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave stripe and dark rogue wave solutions of a generalized Kadomtsev-Petviashvili equation in fluid mechanics. Chin. J. Phys. 60, 440–449 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lonngren, K.E.: Ion acoustic soliton experiments in a plasma. Opt. Quantum Electron. 30, 615–630 (1998)CrossRefGoogle Scholar
  15. 15.
    Tsuchiya, S., Dalfovo, F., Pitaevskii, L.: Solitons in two-dimensional Bose-Einstein condensates. Phys. Rev. A 77, 045601 (2008)CrossRefGoogle Scholar
  16. 16.
    Lan, Z.Z.: Periodic, breather and rogue wave solutions for a generalized (\(3+1\))-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid dynamics. Appl. Math. Lett. 94, 126–132 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Date, E., Jimbo, M., Kashiwara, M.: A new hierarchy of soliton equations of KP-type. Phys. D 4, 343–365 (1982)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hu, C.C., Tian, B., Wu, X.Y., Du, Z., Zhao, X.H.: Lump wave-soliton and rogue wave-soliton interactions for a (\(3+1\))-dimensional B-type Kadomtsev-Petviashvili equation in a fluid. Chin. J. Phys. 56, 2395 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hu, C.C., Tian, B., Wu, X.Y., Yuan, Y.Q., Du, Z.: Mixed lump-kink and rogue wave-kink solutions for a (\(3+1\))-dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics. Eur. Phys. J. Plus 133, 40 (2018)CrossRefGoogle Scholar
  20. 20.
    Asaad, M.G., Ma, W.X.: Pfaffian solutions to a (\(3+1\))-dimensional generalized B-type Kadomtsev-Petviashvili equation and its modified counterpart. Appl. Math. Comput. 218, 5524–5542 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sun, Y., Tian, B., Liu, L.: Rogue waves and lump solitons of the (\(3+ 1\))-dimensional generalized B-type Kadomtsev-Petviashvili equation for water waves. Commun. Theor. Phys. 68, 693 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ma, W.X., Zhu, Z.: Solving the (\(3+ 1\))-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ma, W.X., Fan, E.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61, 950–959 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chabchoub, A., Hoffmann, N., Onorato, M.: Super rogue waves: observation of a higher-order breather in water waves. Phys. Rev. X 2, 011015 (2012)Google Scholar
  25. 25.
    Dudley, J.M., Genty, G., Dias, F.: Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. Opt. Express 17, 21497–21508 (2009)CrossRefGoogle Scholar
  26. 26.
    Du, Z., Tian, B., Chai, H.P., Yuan, Y.Q.: Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrodinger equations in an alpha helical protein. Commun. Nonlinear Sci. Numer. Simulat. 67, 49 (2019)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Du, Z., Tian, B., Chai, H.P., Sun, Y., Zhao, X.H.: Rogue waves for the coupled variable-coefficient fourth-order nonlinear Schrodinger equations in an inhomogeneous optical fiber. Chaos, Solitons Fractals 109, 90 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sun, Y., Tian, B., Liu, L., Wu, X.Y.: Rogue waves for a generalized nonlinear Schrodinger equation with distributed coefficients in a monomode optical fiber. Chaos, Solitons Fractals 107, 266 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Russell, F.M., Archilla, J.F.R., Frutos, F.: Infinite charge mobility in muscovite at 300 K. EPL 120, 46001 (2012)CrossRefGoogle Scholar
  30. 30.
    Akhmediev, N., Soto-Crespo, J.M., Ankiewicz, A.: How to excite a rogue wave. Phys. Rev. A 80, 043818 (2009)CrossRefGoogle Scholar
  31. 31.
    Wazwaz, A.M.: Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations. Nonlinear Dyn. 85, 731–737 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    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
  33. 33.
    Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496–1503 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ma, W.X.: Lump solutions to the Kadomtsev-Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zhang, J., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. ANZIAM J. 25, 16–43 (1983)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026601 (2009)CrossRefGoogle Scholar
  38. 38.
    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)CrossRefGoogle Scholar
  39. 39.
    Lan, Z.Z., Su, J.J.: Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system. Nonlinear Dyn. 96, 2535–2546 (2019)CrossRefGoogle Scholar
  40. 40.
    Ma, Y.L.: Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers. Nonlinear Dyn.
  41. 41.
    Chen, S.S., Tian, B., Sun, Y., Zhang, C.R.: Generalized Darboux transformations, rogue waves, and modulation instability for the coherently coupled nonlinear Schrödinger equations in nonlinear optics. Ann. Phys. (2019). Google Scholar
  42. 42.
    Chen, S.S., Tian, B., Liu, L., Yuan, Y.Q., Zhang, C.R.: Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system. Chaos, Solitons Fractals 118, 337 (2019)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhang, C.R., Tian, B., Wu, X.Y., Yuan, Y.Q., Du, X.X.: Rogue waves and solitons of the coherently coupled nonlinear Schrödinger equations with the positive coherent coupling. Phys. Scr. 93, 095202 (2018)CrossRefGoogle Scholar
  44. 44.
    Zhang, C.R., Tian, B., Liu, L., Chai, H.P., Du, Z.: Vector breathers with the negatively coherent coupling in a weakly birefringent fiber. Wave Motion 84, 68 (2019)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089–1093 (1986)CrossRefzbMATHGoogle Scholar
  46. 46.
    Ma, Y.C.: The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43–58 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Zhang, X.E., Chen, Y.: Deformation rogue wave to the (\(2+1\))-dimensional KdV equation. Nonlinear Dyn. 90, 755–763 (2017)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Zhang, X.E., Chen, Y.: General high-order rogue wave to NLS-Boussinesq equation with the dynamical analysis. Nonlinear Dyn.
  49. 49.
    Zakharov, V.E., Dyachenko, A.I.: About shape of giant breather. Eur. J. Mech. B 29, 127–131 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Zakharov, V.E., Kuznetsov, E.A.: Multi-scale expansions in the theory of systems integrable by the inverse scattering transform. Phys. D 69, 455–463 (1986)CrossRefzbMATHGoogle Scholar
  51. 51.
    Gao, X.Y.: Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation Appl. Math. Lett. 73, 143 (2017)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Gao, X.Y.: Mathematical view with observational/experimental consideration on certain (\(2+1\))-dimensional waves in the cosmic/laboratory dusty plasmas. Appl. Math. Lett. 91, 165 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Dysthe, K.B., Trulsen, K.: Note on breather type solutions of the NLS as models for freak-waves. Phys. Scr. 82, 48 (1999)CrossRefGoogle Scholar
  54. 54.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits. Phys. Rev. E 85, 066601 (2012)CrossRefGoogle Scholar
  55. 55.
    Ablowitz, M.J., Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19, 2180–2186 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  57. 57.
    Ohta, Y., Yang, J.: Rogue waves in the Davey-Stewartson I equation. Phys. Rev. E 86, 036604 (2012)CrossRefGoogle Scholar
  58. 58.
    Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468, 1716–1740 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yuan, Y.Q., Tian, B., Liu, L., Wu, X.Y., Sun, Y.: Solitons for the (\(2+1\))-dimensional Konopelchenko-Dubrovsky equations. J. Math. Anal. Appl. 460, 476 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19, 943–1001 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Zhang, X., Xu, T., Chen, Y.: Hybrid solutions to Mel’nikov system. Nonlinear Dyn.
  62. 62.
    Osborne, A.R.: Classification of homoclinic rogue wave solutions of the nonlinear Schrödinger equation. Nat. Hazard Earth Syst. 2, 897–933 (2014)CrossRefGoogle Scholar
  63. 63.
    Ablowitz, M.J., Herbst, B.M.: On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 50, 339–351 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid DynamicsBeijing University of Aeronautics and AstronauticsBeijingChina

Personalised recommendations