Nonlinear Dynamics

, Volume 95, Issue 1, pp 273–291 | Cite as

Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation

  • Min-Jie Dong
  • Shou-Fu TianEmail author
  • Xue-Wei Yan
  • Tian-Tian Zhang
Original Paper


We consider the classical Boussinesq–Burgers (BB) equation, which describes the propagation of shallow water waves. Based on the truncated painlevé expansion method and consistent Riccati expansion method, we successfully obtain its nonlocal symmetry and Bäcklund transformation. By introducing auxiliary-dependent variables for the nonlocal symmetry, we find the corresponding Lie point symmetries. By considering the consistent tanh expansion method, the interaction solution of soliton–cnoidal wave for the classical BB equation is studied by using the Jacobi elliptic function. The multi-solitary wave solutions are also obtained by introducing a linear combination of N exponential functions. Moreover, the conservation laws of the equation are successfully obtained with a detailed derivation.


The classical Boussinesq–Burgers equation Truncated painlevé expansion Interaction solutions Lie point symmetry Conservation laws 



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 and Practice Program of Education and Teaching Reform of CUMT under Grant No. YJSJG_2018_036, the “Qinglan Engineering project” of Jiangsu Universities, the National Natural Science Foundation of China under Grant No. 11301527, the Fundamental Research Fund for the Central Universities under the Grant No. 2017XKQY101, and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M57 0498 and 2017T100413.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.School of Mathematics and Institute of Mathematical PhysicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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