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Bäcklund transformation, rogue wave solutions and interaction phenomena for a \(\varvec{(3+1)}\)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

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Abstract

Under investigation in this paper is the \((3+1)\)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq (BKP–Boussinesq) equation, which can display the nonlinear dynamics in fluid. By using Bell’s polynomials, we explicitly derive a bilinear equation for the equation via a very natural and effective way. Then, three types of exchange identities of Hirota’s bilinear operators are presented to derive its Bäcklund transformation. Based on that, we construct the traveling wave solutions, kink solitary wave solutions, rational breathers and rogue waves of the equation. Finally, some properties of interaction phenomena are also provided, which can be used to study the domain of lump solutions. It is hoped that our results can be used to enrich the dynamical behavior of the \((3+1)\)-dimensional nonlinear evolution equations.

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References

  1. 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 

  2. 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 

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

    MATH  Google Scholar 

  4. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, L., Zhang, J.H., Wang, Z.Q., Liu, C., Li, M., Qi, F.H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)

    Article  MathSciNet  Google Scholar 

  7. Guo, R., Hao, H.Q.: Breathers and localized solitons for the Hirota–Maxwell–Bloch system on constant backgrounds in erbium doped fibers. Ann. Phys. 344, 10–16 (2014)

    Article  MATH  Google Scholar 

  8. 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 

  9. 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). (22 pp)

    Article  MATH  Google Scholar 

  10. 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  Google Scholar 

  11. Yu, F.J.: Nonautonomous rogue waves and ‘catch’ dynamics for the combined Hirota–LPD equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 34, 142–153 (2016)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    MathSciNet  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MATH  Google Scholar 

  16. Ablowitz, M.J., Clarkson, P.A.: Solitons; Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  17. Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  19. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)

    Book  MATH  Google Scholar 

  20. Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)

    Book  MATH  Google Scholar 

  21. 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)

    Article  MATH  Google Scholar 

  22. Johnson, R.S., Thompson, S.: A solution of the inverse scattering problem for the Kadomtsev–Petviashvili equation by the method of separation of variables. Phys. Lett. 66A(4), 279–281 (1978)

    Article  MathSciNet  Google Scholar 

  23. Wazwaz, A.M.: Two forms of (3+ 1)-dimensional B-type Kadomtsev–Petviashvili equation: multiple soliton solutions. Phys. Scr. 86(3), 035007 (2012)

    Article  MATH  Google Scholar 

  24. Ma, W.X., Zhu, Z.N.: Solving the \((3+1)\)-dimensional generalized KP and BKP equations by the multiple \(exp\)-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Feng, L.L., Tian, S.F., Wang, X.B., Zhang, T.T.: Rogue waves, homoclinic breather waves and soliton waves for the \((2+1)\)-dimensional B-type Kadomtsev–Petviashvili equation. Appl. Math. Lett. 65, 90–97 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Characteristics of the breathers, rogue waves and solitary waves in a generalized \((2+1)\)-dimensional Boussinesq equation. EPL 115, 10002 (2016)

    Article  Google Scholar 

  27. Wang, X.B., Tian, S.F., Yan, H., Zhang, T.T.: On the solitary waves, breather waves and rogue waves to a generalized \((3+1)\)-dimensional Kadomtsev–Petviashvili equation. Comput. Math. Appl. 74, 556–563 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Dynamics of the breathers, rogue waves and solitary waves in the \((2+1)\)-dimensional Ito equation. Appl. Math. Lett. 68, 40–47 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gilson, C.R., Nimmo, J.J.C.: Lump solutions of the BKP equations. Phys. Lett. A 147(8–9), 472–476 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  31. 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)

    Article  MathSciNet  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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 

  34. La, 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 

  35. 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 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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 

  39. 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 

  40. 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 

  41. 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 

  42. 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)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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)

    Article  MathSciNet  MATH  Google Scholar 

  44. 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)

    Article  MATH  Google Scholar 

  45. 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)

    Article  MathSciNet  MATH  Google Scholar 

  46. 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)

    Article  MathSciNet  MATH  Google Scholar 

  47. 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. Taiwan. J. Math. 20, 823–848 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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)

    Article  MathSciNet  MATH  Google Scholar 

  50. 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)

    Article  MathSciNet  MATH  Google Scholar 

  51. 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)

    Article  MathSciNet  MATH  Google Scholar 

  52. 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)

    MathSciNet  Google Scholar 

  53. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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 Nos. 2017XKQY101 and DUT17ZD233, and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M570498 and 2017T100413.

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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

    Xue-Wei Yan, Shou-Fu Tian & Min-Jie Dong

  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

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  1. Xue-Wei Yan
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Correspondence to Li Zou.

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Yan, XW., Tian, SF., Dong, MJ. et al. Bäcklund transformation, rogue wave solutions and interaction phenomena for a \(\varvec{(3+1)}\)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn 92, 709–720 (2018). https://doi.org/10.1007/s11071-018-4085-5

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