Nonlinear Dynamics

, Volume 86, Issue 3, pp 1855–1862 | Cite as

The \(\varvec{(2+1)}\)-dimensional Konopelchenko–Dubrovsky equation: nonlocal symmetries and interaction solutions

  • Bo Ren
  • Xue-Ping Cheng
  • Ji Lin
Original Paper


The nonlocal symmetries for the \((2+1)\)-dimensional Konopelchenko–Dubrovsky equation are obtained with the truncated Painlevé method and the Möbious (conformal) invariant form. The nonlocal symmetries are localized to the Lie point symmetries by introducing auxiliary dependent variables. The finite symmetry transformations are obtained by solving the initial value problem of the prolonged systems. The multi-solitary wave solution is presented with the finite symmetry transformations of a trivial solution. In the meanwhile, symmetry reductions in the enlarged systems are studied by the Lie point symmetry approach. Many explicit interaction solutions between solitons and cnoidal periodic waves are discussed both in analytical and in graphical ways.


Konopelchenko–Dubrovsky equation Nonlocal symmetries Symmetry reduction 



This work was supported by Zhejiang Provincial Natural Science Foundation of China under Grant (Nos. LZ15A050001 and LQ16A010003) and the National Natural Science Foundation of China under Grant (Nos. 11305106 and 11505154)


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute of Nonlinear ScienceShaoxing UniversityShaoxingChina
  2. 2.Department of PhysicsZhejiang Ocean UniversityZhoushanChina
  3. 3.Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang ProvinceZhoushanChina
  4. 4.Institute of Nonlinear PhysicsZhejiang Normal UniversityJinhuaChina

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