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

  • Jin-Jin Mao
  • Shou-Fu Tian
  • Li ZouEmail author
  • Tian-Tian Zhang
  • Xing-Jie Yan
Original Paper


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.


A (3+1)-dimensional B-type Kadomtsev–Petviashvili equation Bilinear formalism Lump solution Lumpoff solution Instanton/rogue wave solution 



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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Jin-Jin Mao
    • 1
  • Shou-Fu Tian
    • 1
  • Li Zou
    • 2
    • 3
    Email author
  • Tian-Tian Zhang
    • 1
  • Xing-Jie Yan
    • 1
  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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