# Approximate symmetries and invariant solutions for the generalizations of the Burgers- Korteweg-de Vries mod

Document Type : Research Article

Authors

1 Department of Mathematics, Payame Noor University, Tehran, Iran

2 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

Abstract

In this paper the generalizations of the Burgers-Korteweg-de Vries model with small parameter derived by Kudryshov et al[ N.A. Kudryashov, D.I. Sinelshchikov. Extended models of non-linear waves in liquid with gas bubbles, International Journal of Non-Linear Mechanics 63 (2014) 31-38] is studied. A comprehensive study on the approximate symmetry analysis of the waves models is presented. First, we obtain approximate symmetry for the equation. Subsequently, in a physical application, using the first-order approximate symmetries, corresponding approximate invariant solutions to the perturbed non-linear models are obtained.

Keywords

20.1001.1.25882953.2019.51.2.17.8

Main Subjects

#### References

[1]      N.A. Kudryashov, D.I. Sinelshchikov. Extended models of non-linear waves in liquid with gas bubbles, International Journal of Non-Linear Mechanics 63 (2014) 31-38.
[2]      N.A. Kudryashov, D.I. Sinelshchikov. Periodic structures described by the perturbed Burgers-Korteweg-de Vries equation, International Journal of Non-Linear Mechanics 72 (2015) 16-22.
[3]      G.F. Jefferson, J. Carminati. ASP: Automated symbolic computation of approximate symmetries of differential equations. Computer Physics Communications 184 (2013) 1045-1063.
[4]      V.A. Baikov, R.K. Gazizov, N.Kh. Ibragimov. Perturbation methods in group analysis. J. Sov. Math 1991;55:1450-90. grade, Acta Mechanica Sinica 24(6), 661-670, 2008.
[5]      V.A. Baikov, R.K. Gazizov, N.H. Ibragimov. Approximate symmetries of equations with a small parameter. Mat Sb 1988;136:435-50.
[6]      M. Pakdemirli, M. Yurusoy, T. Dolapc. Comparison of approximate symmetry methods for differential equations. Acta Appl Math 2004;80:243-71.
[7]      N.H. Ibragimov, Selected works, vol. III, 2008, 317p. ISBN 978-91-7295-992-7.
[8]      N.H. Ibragimov, V.F. Kovalev. Approximate and Renormgroup Symmetries, Higher Education Press, Beijing and SpringerVerlag GmbH Berlin Heidelberg, 2009.
[9]      R.K. Gazizov. Lie Algebras of Approximate Symmetries, Nonlinear Mathematical Physics 1996, V.3, N 1-2, 96-101.
[10]   N.A. Kudryashov, D.I. Sinelshchikov. Special solutions of a high-order equation for waves in a liquid with gas bubbles, Regul. Chaotic Dyn. 19 (2014), 576-585.
[11]   R. Dastranj, M. Nadjafikhah. Symmetry analysis and conservation laws for description of waves in bubbly liquid, International Journal of Non-Linear Mechanics, 67 (2014) 48-51.