SIGMA 2 (2006), 040, 11 pages      math-ph/0604021      https://doi.org/10.3842/SIGMA.2006.040

Nonclassical Approximate Symmetries of Evolution Equations with a Small Parameter

Svetlana Kordyukova
Department of Mathematics, Ufa State Aviation Technical University, 12 K. Marx Str., Ufa, 450000 Russia

Received November 30, 2005, in final form March 17, 2006; Published online April 10, 2006

Abstract
We introduce a method of approximate nonclassical Lie-Bäcklund symmetries for partial differential equations with a small parameter and discuss applications of this method to finding of approximate solutions both integrable and nonintegrable equations.

Key words: nonclassical Lie-Bäcklund symmetries; approximate symmetry; conditional-invariant solution.

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References

1. Baikov V.A., Gazizov R.K., Ibragimov N.Kh., Perturbation methods in group analysis, Current Problems in Mathematics. Newest Results, Vol. 34, Moscow, VINITI, 1989, 85-147 (English transl.: J. Soviet Math., 1991, V.55, 1450-1490).
2. Bluman G.W., Cole J.D., The general similarity solutions of the heat equation, J. Math. Mech., 1969, V.18, 1025-1042.
3. Olver P.J., Rosenau P., The construction of special solutions to partial differential equation, Phys. Lett. A, 1986, V.114, 107-112.
4. Clarkson P.A., Kruskal M., New similarity reductions of the Boussinesq equation, J. Math. Phys., 1989, V.30, 2201-2213.
5. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear equations of mathematical physics, Kyiv, Naukova Dumka, 1989 (English transl.: Dordrecht, Kluwer, 1993).
6. Fokas A.S., Liu Q.M., Generalized conditional symmetries and exact solutions of non-integrable equations, Teor. Mat. Fiz., 1994, V.99, 263-277 (English transl.: Theor. Math. Phys., 1994, V.99, 571-582).
7. Zhdanov R.Z., Conditional Lie-Bäcklund symmetry and reduction of evolution equations, J. Phys. A: Math. Gen., 1995, V.28, 3841-3850.
8. Mahomed F.M., Qu C., Approximate conditional symmetries for partial differential equations, J. Phys. A: Math. Gen., 2000, V.33, 343-356.
9. Kara A.F., Mahomed F.M., Qu C., Approximate potential symmetries for partial differential equations, J. Phys. A: Math. Gen., 2000, V.33, 6601-6613.
10. Olver P.J., Vorob'ev E.M., Nonclassical and conditional symmetries, in CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, Editor N.H. Ibragimov, Boca Raton, Florida, CRC Press, 1996, 291-328.
11. Sokolov V.V., Shabat A.B., Classification of integrable evolution equations, in Mathematical Physics Reviews, Vol. 4, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 4, Chur, Harwood Academic Publ., 1984, 221-280.
12. Baikov V.A., Kordyukova S.A., Approximate symmetries of the Boussinesq equation, Quaest. Math., 2003, V.26, 1-14.
13. Baikov V.A., Gazizov R.K., Ibragimov N.Kh., Approximate transformation groups and deformations of symmetry Lie algebras, Chapter 2, in CRC Handbook of Lie Group Analysis of Differential Equation, Vol. 3, Editor N.H. Ibragimov, Boca Raton, Florida, CRC Press, 1996.