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Nonlinear Dynamics

, Volume 92, Issue 2, pp 709–720 | Cite as

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

  • Xue-Wei Yan
  • Shou-Fu Tian
  • Min-Jie Dong
  • Li Zou
Original Paper

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.

Keywords

BKP–Boussinesq equation Bäcklund transformation Bell’s polynomial Rogue waves Traveling waves Kink solitary waves Interaction phenomena 

Notes

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.

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.

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© 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

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