Nonlinear Dynamics

, Volume 91, Issue 4, pp 2593–2605 | Cite as

Families of exact solutions of a new extended \(\varvec{(2+1)}\)-dimensional Boussinesq equation

  • Yulei Cao
  • Jingsong He
  • Dumitru Mihalache
Original Paper


A new variant of the \((2+1)\)-dimensional [\((2+1)d\)] Boussinesq equation was recently introduced by Zhu [Line soliton and rational solutions to (2+1)-dimensional Boussinesq equation by Dbar problem, 2017. arXiv:1704.02779v2; see eq. (3)]. First, we derive in this paper the one-soliton solutions of both bright and dark types for the extended \((2+1)d\) Boussinesq equation by using the traveling wave method. Second, N-soliton, breather, and rational solutions are obtained by using the Hirota bilinear method and the long-wave limit. Nonsingular rational solutions of two types were obtained analytically, namely (i) rogue wave solutions having the form of W-shaped lines waves and (ii) lump-type solutions. Two generic types of semi-rational solutions were also put forward. The obtained semi-rational solutions are as follows: (iii) a hybrid of a first-order lump and a bright one-soliton solution and (iv) a hybrid of a first-order lump and a first-order breather.


\((2+1)\)-dimensional Boussinesq Solitons Breathers Rogue waves Semi-rational solutions Bilinear method 



This work is supported by the NSF of China under Grant No. 11671219 and the K.C. Wong Magna Fund in Ningbo University. We thank other members in our group at Ningbo University for many useful discussions on the paper.

Compliance with ethical standards

Conflict of interest

Authors declare that they have no conflict of interests.


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Authors and Affiliations

  1. 1.Department of MathematicsNingbo UniversityZhejiangPeople’s Republic of China
  2. 2.Horia Hulubei National Institute for Physics and Nuclear EngineeringMagureleRomania

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