Симетрія Лі–Беклунда, редукція і розв’язки нелінійних еволюційних рівнянь

W.  Rzeszut; I. M. Tsyfra; V. A.  Vladimirov

doi:10.37863/umzh.v74i3.7007

W. Rzeszut AGH University of Science and Technology, Krakow, Poland
I. M. Tsyfra AGH University of Science and Technology, Krakow, Poland; The Institute of Geophysics of the National Academy of Sciences of Ukraine
V. A. Vladimirov AGH University of Science and Technology, Krakow, Poland

DOI: https://doi.org/10.37863/umzh.v74i3.7007

Keywords: Lie-Backlund symmetry operators, ordinary differential equations, partial differential equations, reduction, invariant solutions

Abstract

UDC 517.9

We study the symmetry reduction of nonlinear partial differential equations which are used for describing diffusion processes in a nonhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary differential equations.
The ansatzes are constructed by using operators of Lie–Backlund symmetry of the third order ordinary differential equation. The method gives a possibility to find solutions which can not be obtained by virtue of the classical Lie method. Such solutions were constructed for nonlinear diffusion equations which are invariant with respect to one-parameter, two-parameter, and three-parameter Lie groups of point transformations.

References

P. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, New York (1993), https://doi.org/10.1007/978-1-4612-4350-2

G. Bluman, J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech., 18, № 11, 1025 – 1042 (1969).

P. J. Olver, P. Rosenau, The construction of special solutions to partial differential equations, Phys. Lett. A, 114, № 3, 107 – 112 (1986), https://doi.org/10.1016/0375-9601(86)90534-7

P. J. Olver, P. Rosenau, Group-invariant solutions of differential equations, SIAM J. Appl. Math., 47, № 2, 263 – 278 (1987), https://doi.org/10.1137/0147018

W. I. Fushchych, I. M. Tsyfra, On a reduction and solutions of nonlinear wave equations with broken symmetry, J. Phys. A., 20, № 2, L45 – L48 (1987), http://stacks.iop.org/0305-4470/20/L45

A. S. Fokas, Q. M. Liu, Nonlinear interaction of traveling waves of non-integrable equations, Phys. Rev. Letters, 72, № 21, 3293 – 3296 (1994), https://doi.org/10.1103/PhysRevLett.72.3293

R. Z. Zhdanov, Conditional Lie-Backlund symmetry and reduction of evolution equations , J. Phys. A: Math. Gen., 28, 3841 – 3850 (1995), http://stacks.iop.org/0305-4470/28/3841

C. Z. Qu, Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math., 62, № 3, 283 – 302 (1999), https://doi.org/10.1093/imamat/62.3.283

S. R. Svirshchevskii, Lie-Backlund symmetries of linear ODEs and generalized separation of variables in nonlinear equations, Phys. Lett. A, 199, № 5-6, 344 – 349 (1995), https://doi.org/10.1016/0375-9601(95)00136-Q

I. M. Tsyfra, Symmetry reduction of nonlinear differential equations, Proc. Ins. Math., Kiev,50, 266 – 270 (2004).

M. Kunzinger, R. O. Popovych, Generalized conditional symmetry of evolution equations, J. Math. Anal. Appl., 379, № 1, 444 – 460 (2011), https://doi.org/10.1016/j.jmaa.2011.01.027

I. M. Tsyfra, Conditional symmetry reduction and invariant solutions of nonlinear wave equations, Proc. Ins. Math., 43, 229 – 233 (2002).

I. M. Tsyfra, W. Rzeszut, Lie-Backlund symmetry reduction of nonlinear and non-evolution equations, Proc. Ins. Math., 16, № 1, 174 – 180 (2019).

I. Tsyfra, T. Czyżycki, Nonpoint symmetry and reduction of nonlinear evolution and wave type equations, Abstract and Applied Anal., 2015, Article ID 181275 (2015), https://doi.org/10.1155/2015/181275

Lie-Backlund symmetry, reduction and solutions of nonlinear evolution equations

Abstract

References