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Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients. (English) Zbl 1221.35338
Summary: This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.

35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
Full Text: DOI
[1] Antonova, M.; Biswas, A., Adiabatic parameter dynamics of perturbed solitary waves, Commun. nonlinear sci. numer. simul., 14, 3, 734-748, (2009) · Zbl 1221.35321
[2] Biswas, A., Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion, Commun. nonlinear sci. numer. simul., 14, 3503-3506, (2009) · Zbl 1221.35306
[3] Xiao-Yan, T.; Fei, H.; Sen-Yue, L., Variable coefficient KdV equation and the analytical diagnosis of a dipole blocking life cycle, Chinese phys. lett., 23, 887-890, (2006)
[4] Wazwaz, A.M., New sets of solitary wave solutions to the KdV, mkdv and generalized KdV equations, Commun. nonlinear sci. numer. simul., 13, 2, 331-339, (2008) · Zbl 1131.35385
[5] Triki, H.; Wazwaz, A.M., Soliton solutions for (2+1)-dimensional and (3+1)-dimensional K(m,n) equations, Appl. math. comput., (2009)
[6] Lie, S., On integration of a class of linear partial differential equations by means of definite integrals, Arch. math., VI, 3, 328-368, (1881) · JFM 13.0298.01
[7] Pakdemirli, M.; Sahin, A.Z., Group classification of fin equation with variable thermal properties, Int. J. eng. sci., 42, 1875-1889, (2004) · Zbl 1211.35141
[8] Sophocleous, C., Further transformation properties of generalized inhomogeneous nonlinear diffusion equations with variable coeffcients, Physica A, 345, 457-471, (2005)
[9] Liu, H.; Li, J.; Liu, L., Lie group classifications and exact solutions for two variable-coefficient equations, Appl. math. comp., 215, 2927-2935, (2009) · Zbl 1232.35173
[10] Nadjafikah, M.; Bakhshandeh-Chamazkoti, R., Symmetry group classification for burgers’ equation, Commun. nonlinear sci. numer. simul., 152, 2303-2310, (2010) · Zbl 1222.35195
[11] Senthilkumaran, M.; Pandiaraja, D.; Vaganan, B.M., New exact explicit solutions of the generalized KdV equations, Appl. math. comput., 202, 2, 693-699, (2008) · Zbl 1158.35420
[12] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York · Zbl 0698.35001
[13] N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1-3, CRC Press, Boca Raton, FL, 1994-1996.
[14] Olver, P.J., Applications of Lie groups to differential equations, (1993), Springer New York · Zbl 0785.58003
[15] Ovsiannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002
[16] ()
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