Nonlinear Dynamics

, Volume 94, Issue 4, pp 2547–2561 | Cite as

On Lie symmetries and soliton solutions of \((2+1)\)-dimensional Bogoyavlenskii equations

  • Mukesh KumarEmail author
  • Dig Vijay Tanwar
  • Raj Kumar
Original Paper


The present article is devoted to find some invariant solutions of the \((2+1)\)-dimensional Bogoyavlenskii equations using similarity transformations method. The system describes \((2+1)\)-dimensional interaction of a Riemann wave propagating along y-axis with a long wave along x-axis. All possible vector fields, commutative relations and symmetry reductions are obtained by using invariance property of Lie group. Meanwhile, the method reduces the number of independent variables by one, which leads to the reduction of Bogoyavlenskii equations into a system of ordinary differential equations. The system so obtained is solved under some parametric restrictions and provides invariant solutions. The derived solutions are much efficient to explain the several physical properties depending upon various existing arbitrary constants and functions. Moreover, some of them are more general than previously established results (Peng and Shen in Pramana 67:449–456, 2006; Malik et al. in Comput Math Appl 64:2850–2859, 2012; Zahran and Khater in Appl Math Model 40:1769–1775, 2016; Zayed and Al-Nowehy in Opt Quant Electron 49(359):1–23, 2017). In order to provide rich physical structures, the solutions are supplemented by numerical simulation, which yield some positons, negatons, kinks, wavefront, multisoliton and asymptotic nature.


Bogoyavlenskii equations Lie group theory Symmetry reductions Invariant solutions 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsMotilal Nehru National Institute of Technology AllahabadAllahabadIndia
  2. 2.Department of MathematicsV. B. S. Purvanchal UniversityJaunpurIndia

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