Geodesic
Classical and Nonclassical Symmetries of Nonlinear Differential Equation for Describing Waves in a Liquid with Gas Bubbles
Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 1, pp. 45-52.

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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.
Keywords: nonlinear waves in a liquid with gas bubbles, classical symmetries, nonclassical symmetries
Mots-clés : exact solutions. 
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N. A. Kudryashov; D. I. Sinelshchikov. Classical and Nonclassical Symmetries of Nonlinear Differential Equation for Describing Waves in a Liquid with Gas Bubbles. Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 1, pp. 45-52. http://geodesic.mathdoc.fr/item/MAIS_2014_21_1_a2/

[1] R. I. Nigmatulin, Dynamics of Multiphase Media, v. 2, Taylor Francis, New York, 1990, 388 pp.

[2] V. E. Nakoryakov, B. G. Pokusaev, I. R. Shreiber, Wave Propagation in Gas-Liquid Media, CRC Press, Boca Raton, 1993, 240 pp.

[3] N. A. Kudryashov, D. I. Sinelshchikov, “An extended equation for the description of nonlinear waves in a liquid with gas bubbles”, Wave Motion, 50:3 (2013), 351–362 | DOI | MR

[4] J. Weiss, M. Tabor, G. Carnevale, “The Painleve property for partial differential equations”, J. Math. Phys., 24 (1983), 522–526 | DOI | MR | Zbl

[5] N. A. Kudryashov, “On types of nonlinear nonintegrable equations with exact solutions”, Phys. Lett. A, 155:4–5 (1991), 269–275 | DOI | MR

[6] N. A. Kudryashov, “Singular manifold equations and exact solutions for some nonlinear partial differential equations”, Phys. Lett. A, 182:4–6 (1993), 356–362 | DOI | MR

[7] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Waltham, 1982, 432 pp. | MR | Zbl

[8] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993, 513 pp. | MR | Zbl

[9] N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Mathematics and its Applications, Springer, New York, 2001, 396 pp.

[10] G. W. Bluman, J. D. Cole, “The general similarity solution of the heat equation”, J. Math. Mech., 18:11 (1969), 1025–1042 | MR | Zbl

[11] R. Z. Zhdanov, I. M. Tsyfra, R. O. Popovych, “A Precise Definition of Reduction of Partial Differential Equations”, J. Math. Anal. Appl., 238:1 (1999), 101–123 | DOI | MR | Zbl

[12] M. Kunzinger, R. O. Popovych, “Singular reduction operators in two dimensions”, J. Phys. A Math. Theor., 41:50 (2008), 505201 | DOI | MR | Zbl