We study the problem of symmetry reduction of nonlinear partial differential equations used to describe diffusion processes in an inhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary differential equations. These ansatzes are constructed by using the operators of Lie–Bäcklund symmetry of the third-order ordinary differential equations. The proposed method enables us to find solutions that cannot be obtained by using the classical Lie method. These solutions are constructed for nonlinear diffusion equations invariant under one-, two-, and three-parameter Lie groups of point transformations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1993).
G. Bluman and J. D. Cole, “The general similarity solution of the heat equation,” J. Math. Mech., 18, No.11, 1025–1042 (1969).
P. J. Olver and P. Rosenau, “The construction of special solutions to partial differential equations,” Phys. Lett., 114A, No. 3, 107–112 (1986).
P. J. Olver and P. Rosenau, “Group-invariant solutions of differential equations,” SIAM J. Appl. Math., 47, No. 2, 263–278 (1987).
W. I. Fushchych and I. M. Tsyfra, “On a reduction and solutions of nonlinear wave equations with broken symmetry,” J. Phys. A., 20, No. 2, L45–L48 (1987).
A. S. Fokas and Q. M. Liu, “Nonlinear interaction of traveling waves of non-integrable equations,” Phys. Rev. Lett., 72, No. 21, 3293–3296 (1994).
R. Z. Zhdanov, “Conditional Lie–Bäcklund symmetry and reduction of evolution equations,” J. Phys. A: Math. Gen., 28, 3841–3850 (1995).
C. Z. Qu, “Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method,” IMA J. Appl. Math., 62, 283–302 (1999).
S. R. Svirshchevskii, “Lie–Bäcklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations,” Phys. Lett. A, 199, 344–349 (1995).
I. M. Tsyfra, “Symmetry reduction of nonlinear differential equations,” Proc. Inst. Math., 50, 266–270 (2004).
M. Kunzinger and R. O. Popovych, “Generalized conditional symmetry of evolution equations,” J. Math. Anal. Appl., 379, No. 1, 444–460 (2011).
I. M. Tsyfra, “Conditional symmetry reduction and invariant solutions of nonlinear wave equations,” Proc. Inst. Math., 43, 229–233 (2002).
I. M. Tsyfra and W. Rzeszut, “Lie–Backlund symmetry reduction of nonlinear and non-evolution equations,” Proc. Inst. Math., 16, No. 1, 174–180 (2019).
I. Tsyfra and T. Czyzycki, “Nonpoint symmetry and reduction of nonlinear evolution and wave type equations,” Abstr. Applied Anal., 2015, Article ID 181275 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 3, pp. 342–350, March, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i3.7007.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rzeszut, W., Tsyfra, I.M. & Vladimirov, V.A. Lie–Bäcklund Symmetry, Reduction, and Solutions of Nonlinear Evolutionary Equations. Ukr Math J 74, 385–394 (2022). https://doi.org/10.1007/s11253-022-02070-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02070-w