Classification of Lie reductions of generalized Kawahara equations with variable coefficients




Lie symmetries, reduction method, Kawahara equations, group classification, exact solutions


A class of generalized Kawahara equations with time-dependent coefficients is studied from Lie symmetry point of view. A classification of Lie reductions of equations from this class has been carried out. For each case of Lie symmetry extension, the type of the maximal invariance algebra of the corresponding Kawahara equation is determined, and the respective optimal system of one-dimensional subalgebras is found, which are further used to construct Lie ansatzes. Lie reductions of Kawahara equations to ordinary differential equations are performed, some exact Lie invariant solutions are also constructed.


Download data is not yet available.


Kawahara, T. (1972). Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan, 33, pp. 260-271.

Kuriksha, O., Pošta, S. & Vaneeva, O. (2014). Group classification of variable coefficient generalized Kawahara equations. J. Phys. A: Math. Theor., 47, 045201, 19 p.

Kaur, L. & Gupta, R. K. (2013). Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized (G'/G)-expansion method. Math. Meth. Appl. Sci., 36, No. 5, pp. 584-600.

Gandarias, M. L., Rosa, M., Recio, E. & Anco, S. (2017). Conservation laws and symmetries of a generalized Kawahara equation. AIP Conf. Proc., 1836, 020072, 6 p.

Vaneeva, O. O. & Zhalij, A. Yu. (2020). Lie symmetries of generalized Kawahara equations. Dopov. Nac. akad. nauk Ukr., No. 12, pp. 3-10 (in Ukrainian).

Vaneeva, O. O., Bihlo, A. & Popovych, R. O. (2020). Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations. Commun. Nonlinear Sci. Numer. Simulat., 91, 105419, 28 p.

Olver, P. J. (1993). Applications of Lie groups to differential equations. 2nd edn. New York: Springer.

Patera, J. & Winternitz, P. (1977). Subalgebras of real three- and four-dimensional Lie algebras. J. Math. Phys., 18, pp. 1449-1455.

Vaneeva, O. O., Sophocleous, C. & Leach, P. G. L. (2015). Lie symmetries of generalized Burgers equations: ap pli cation to boundary-value problems. J. Eng. Math., 91, No. 1, pp. 165-176.

Vaneeva, O. O., Papanicolaou, N. C., Christou, M. A. & Sophocleous, C. (2014). Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries. Commun. Nonlinear Sci. Numer. Simulat., 19, No. 9, pp. 3074-3085.



How to Cite

Vaneeva О. ., Zhalij О. ., & Magda О. . (2022). Classification of Lie reductions of generalized Kawahara equations with variable coefficients. Reports of the National Academy of Sciences of Ukraine, (6), 3–9.