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

Original Paper
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Abstract

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.

Keywords

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

Notes

Acknowledgements

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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© Springer Science+Business Media B.V., part of Springer Nature 2018

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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