Abstract
Lie group analysis is applied to carry out the similarity reductions of the \((3+1)\)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. We obtain generators of infinitesimal transformations of the CBS equation and each of these generators depend on various parameters which give us a set of Lie algebras. For each of these Lie algebras, Lie symmetry method reduces the \((3+1)\)-dimensional CBS equation into a new \((2+1)\)-dimensional partial differential equation and to an ordinary differential equation. In addition, we obtain commutator table of Lie brackets and symmetry groups for the CBS equation. Finally, we obtain closed-form solutions of the CBS equation by using the invariance property of Lie group transformations.
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References
He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708 (2006)
Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Miura, M.R.: Bcklund Transformation. Springer, Berlin (1978)
Wang, M., Li, X., Zhang, J.: The \((\frac{G^{\prime }}{G})\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)
Bluman, G.W., Cole, J.D.: Similarity Methods for Differential Equations. Springer, New York (1974)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Sahoo, S., Garai, G., Saha Ray, S.: Lie symmetry analysis for similarity reduction and exact solutions of modified KdV–Zakharov–Kuznetsov equation. Nonlinear Dyn. 87, 1995–2000 (2017)
Johnpillai, A.G., Kara, A.H., Biswas, A.: Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation. Appl. Math. Lett. 26, 376–381 (2013)
Wang, G., et al.: Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation. Chaos Solitons Fractals 86, 8–15 (2016)
Jafari, H., Kadkhoda, N., Baleanu, D.: Fractional Lie group method of the time-fractional Boussinesq equation. Nonlinear Dyn. 81, 1569–1574 (2015)
Bogoyavlenski, O. I.: Math. USSR-Izv. 35, 245–248 (1990); translated from Izv. Akad. Nauk SSSR Ser. Mat. 53, 907–910 (1989)
Bogoyavlenski, O. I.: Math. USSR-Izv. 36, 129–137 (1991); translated from Izv. Akad. Nauk SSSR Ser. Mat. 54, 123–131 (1990)
Schiff, J.: Integrability of Chern-Simons-Higgs vortex equations and a reduction of the self-dual Yang-Mills equations to three dimensions, In: Painlevé transcendents (Sainte-Adèle, PQ, 1990), 393–405, NATO Adv. Sci. Inst. Ser. B Phys., 278, Plenum, New York
Toda, K., Yu, S.J.: The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in \((2+1)\) dimensions. J. Math. Phys. 41, 4747–4751 (2000)
Kobayashi, T., Toda, K.: The Painlevé test and reducibility to the canonical forms for higher-dimensional soliton equations with variable-coefficients. SIGMA Symmetry Integrability Geom. Methods Appl. 2, 063, 10 pp (2006)
Bruzon, M. S., et al.: Theoret. Math. Phys. 137, 1378–1389 (2003); translated from Teoret. Mat. Fiz. 137, 27–39 (2003)
Li, S., Li, Y., Zhang, B.: Some singular solutions and their limit forms for generalized Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn. 85, 1665–1677 (2016)
Wang, J., Yang, X.: Quasi-periodic wave solutions for the \((2+1)\)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation. Nonlinear Anal. 75, 2256–2261 (2012)
Wazwaz, A.M.: Multiple-soliton solutions for the Calogero-Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Appl. Math. Comput. 203, 592–597 (2008)
Moatimid, G.M., et al.: Exact solutions for Calogero–Bogoyavlenskii–Schiff equation using symmetry method. Appl. Math. Comput. 220, 455–462 (2013)
Zhang, H.P., Chen, Y., Li, B.: Infinitely many symmetries and symmetry reduction of a \((2+1)\)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. Acta Phys. Sinica 58, 7393–7396 (2009)
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The authors would like to thank the anonymous referees for their extensive comments on the revision of the manuscript which really improved the quality of the paper. The first author expresses her gratitude to the University Grants Commission, New Delhi, India, for financial support to carry out the above work.
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Jadaun, V., Kumar, S. Lie symmetry analysis and invariant solutions of \(\varvec{(3+1)}\)-dimensional Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn 93, 349–360 (2018). https://doi.org/10.1007/s11071-018-4196-z
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DOI: https://doi.org/10.1007/s11071-018-4196-z