Lie symmetry analysis, complex and singular solutions of (\(2+1\))-dimensional combined MCBS–nMCBS equation

  • Sachin KumarEmail author
  • Dharmendra Kumar


In this work, we study Lie symmetry analysis for integrable equation of combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative order (MCBS–nMCBS) that is one of the applications of symmetries. Symmetry analysis and similarity reduction of the (\(2+1\))-dimensional combined MCBS–nMCBS equation are investigated by using the invariance of the equations under the Lie group of transformations. We have found Lie point symmetries of this equation and then we reduced combined MCBS–nMCBS equation to ordinary differential equation through two successive reductions. The same integrable equation was constructed and solved by Wazwaz (Nonlinear Dyn 91:877–883, 2018) using the tanh/coth method. However, we find out the complex and singular solutions using Lie group of transformation method. Some of the derived solutions are analyzed graphically. Eventually, we also obtained conservation laws of the equation with corresponding Lie symmetry.


(\(2+1\))-Dimensional combined MCBS–nMCBS equation Lie point symmetries Invariant solution Optimal system Conservation laws 



The authors would like to thank the anonymous referees and editor for their extensive comments on the revision of the manuscript. This helped to improve the quality of the paper. The second author sincerely and genuinely thanks SGTB Khalsa College, University of Delhi for financial support.

Compliance with ethical standards

Conflicts of interest

The author declares that there is no conflict of interest regarding the publication of this paper.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of DelhiDelhiIndia
  2. 2.Department of Mathematics, SGTB Khalsa CollegeUniversity of DelhiDelhiIndia

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