Nonlinear Dynamics

, Volume 92, Issue 2, pp 221–234 | Cite as

Nonlocal symmetry and similarity reductions for a \(\varvec{(2+1)}\)-dimensional Korteweg–de Vries equation

  • Lili Huang
  • Yong Chen
Original Paper


Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.


Nonlocal symmetry (2 + 1)-dimensional Korteweg–de Vries equation Similarity reduction Interaction solutions 



The authors are very thankful to Lou S Y for his constructive help. The work is supported by the National Natural Science Foundation of China (Grant Nos. 11435005 and 11675054), Outstanding Doctoral Dissertation Cultivation Plan of Action (Grant No. YB2016039), Global Change Research Program of China (Grant No. 2015CB953904) and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213).


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Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina
  2. 2.MOE International Joint Lab of Trustworthy SoftwareEast China Normal UniversityShanghaiChina

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