Lie symmetry analysis of Kudryashov-Sinelshchikov equation. (English) Zbl 1236.35155

Summary: The Lie symmetry method is performed for the fifth-order nonlinear evolution Kudryashov-Sinelshchikov equation. We find one- and two-dimensional optimal systems of Lie subalgebras. Furthermore, the preliminary classification of its group-invariant solutions is investigated.


35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
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[1] S. Lie, Theories der Tranformationgruppen, Dritter und Letzter Abschnitt, Teubner, Leipzig, Germany, 1893.
[2] P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer, New York, NY, USA, 2nd edition, 1993. · Zbl 0785.58003
[3] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81, Springer, New York, NY, USA, 1989. · Zbl 0698.35001
[4] G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations, Springer, New York, NY, USA, 1974. · Zbl 0292.35001
[5] N. A. Kudryashov and D. I. Sinelshchikov, “Nonlinear evolution equation for describing waves in a viscoelastic tube,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2390-2396, 2011. · Zbl 1221.74044
[6] N. A. Kudryashov and M. V. Demina, “Traveling wave solutions of the generalized nonlinear evolution equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 551-557, 2009. · Zbl 1170.35514
[7] Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York, NY, USA, 1993.
[8] N. A. Kudryashov and I. L. Chernyavskii, “Nonlinear waves in fluid flow through a viscoelastic tube,” Fluid Dynamics, vol. 41, no. 1, pp. 49-62, 2006. · Zbl 1198.76167
[9] C. G. Caro, T. J. Pedly, R. C. Schroter, and W. A. Seed, The Mechanics of the Circulation, Oxford University Press, Oxford, UK, 1978. · Zbl 1234.93001
[10] M. Nadjafikhah, “Lie symmetries of inviscid Burgers’ equation,” Advances in Applied Clifford Algebras, vol. 19, no. 1, pp. 101-112, 2009. · Zbl 1173.35699
[11] S. V. Khabirov, “Classification of three-dimensional Lie algebras in R3 and their second-order differential invariants,” Lobachevskii Journal of Mathematics, vol. 31, no. 2, pp. 152-156, 2010. · Zbl 1267.17023
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