Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 515–527 | Cite as

Painlevé analysis, complete Lie group classifications and exact solutions to the time-dependent coefficients Gardner types of equations

  • Hanze Liu
  • Jibin Li
Original Paper


This paper is concerned with the variable-coefficient Gardner (vc-Gardner) types of equations, which arise in fluid dynamics, nonlinear lattice and plasma physics. As its special cases, the generalized cylindrical KdV types of equations are considered simultaneously. By using the combination of Painlevé analysis and Lie group classification method, the integrable conditions, Bäcklund transformations and complete group classifications of the vc-Gardner types of equations are obtained. Then, the exact solutions generated from the Painlevé analysis and symmetry reductions are investigated.


Painlevé analysis Integrable condition Bäcklund transformation Complete group classification Symmetry reduction Exact solution 

Mathematics Subject Classification

37K10 37L20 47G30 



The authors are grateful to the editors and anonymous reviewers for their valuable comments and suggestions.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesLiaocheng UniversityLiaochengChina
  2. 2.Department of MathematicsBinzhou UniversityBinzhouChina
  3. 3.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  4. 4.School of SciencesKunming University of Science and TechnologyKunmingChina

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