Classical and Nonclassical Symmetries of Nonlinear Differential Equation for Describing Waves in a Liquid with Gas Bubbles
https://doi.org/10.18255/1818-1015-2014-1-45-52
Abstract
A nonlinear differential equation is considered for describing nonlinear waves in a liquid with gas bubbles. Classical and nonclassical symmetries of this equation are investigated. It is shown that the considered equation admits transformations in space and time. At a certain condition on parameters, this equation also admits a group of Galilean transformations. The method by Bluman and Cole is used for finding nonclassical symmetries admitted by the studied equation. Both regular and singular cases of nonclassical symmetries are considered. Five families of nonclassical symmetries admitted by this equation are constructed. Symmetry reductions corresponding to these families of generators are obtained. Exact solutions of these symmetry reductions are constructed. These solutions are expressed via rational, exponential, trigonometric and special functions.
About the Authors
N. A. Kudryashov
National Research Nuclear University MEPhI
Russian Federation
Russian Federation
доктор физ.-мат. наук, профессор, зав. кафедрой,
Kashirskoe shosse, 31, Moscow, 115409, Russia
D. I. Sinelshchikov
National Research Nuclear University MEPhI
Russian Federation
Russian Federation
кандидат физ.-мат. наук, ст. преподаватель,
Kashirskoe shosse, 31, Moscow, 115409, Russia
References
1. Nigmatulin R.I. Dynamics of Multiphase Media, Part 2. New York: Taylor & Francis, 1990. P. 388.
2. Nakoryakov V.E., Pokusaev B.G., Shreiber I.R. Wave Propagation in Gas-Liquid Media. Boca Raton: CRC Press, 1993. P. 240.
3. Kudryashov N.A., Sinelshchikov D.I. An extended equation for the description of nonlinear waves in a liquid with gas bubbles // Wave Motion. 2013. Vol. 50, № 3. P. 351–362.
4. Weiss J., Tabor M., Carnevale G. The Painleve property for partial differential equations // J. Math. Phys. 1983. Vol. 24. P. 522–526.
5. Kudryashov N.A. On types of nonlinear nonintegrable equations with exact solutions // Phys. Lett. A. 1991. Vol. 155, № 4-5. P. 269–275.
6. Kudryashov N.A. Singular manifold equations and exact solutions for some nonlinear partial differential equations // Phys. Lett. A. 1993. Vol. 182, № 4–6. P. 356–362.
7. Ovsiannikov L.V. Group Analysis of Differential Equations. Waltham: Academic Press, 1982. P. 432.
8. Olver P.J. Applications of Lie Groups to Differential Equations. New York: Springer, 1993. P. 513.
9. Ibragimov N.H. Transformation Groups Applied to Mathematical Physics (Mathematics and its Applications). New York: Springer, 2001. P. 396.
10. Bluman G.W., Cole J.D. The general similarity solution of the heat equation // J. Math. Mech. 1969. Vol. 18, № 11. P. 1025–1042.
11. Zhdanov R.Z., Tsyfra I.M., Popovych R.O. A Precise Definition of Reduction of Partial Differential Equations // J. Math. Anal. Appl. 1999. Vol. 238, № 1. P. 101–123.
12. Kunzinger M., Popovych R.O. Singular reduction operators in two dimensions // J. Phys. A Math. Theor. 2008. Vol. 41, № 50. P. 505201.
Review
For citations:
Kudryashov N.A., Sinelshchikov D.I. Classical and Nonclassical Symmetries of Nonlinear Differential Equation for Describing Waves in a Liquid with Gas Bubbles. Modeling and Analysis of Information Systems. 2014;21(1):45-52. (In Russ.) https://doi.org/10.18255/1818-1015-2014-1-45-52
Views:
922
Email this article 