## Approximate symmetry and exact solutions of the singularly perturbed Boussinesq equation.(English)Zbl 07261233

Summary: In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the singularly perturbed Boussinesq equation. The tanh-function method, is employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with small parameter which is effectively obtained by the proposed method.

### MSC:

 35B06 Symmetries, invariants, etc. in context of PDEs 76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics 58J70 Invariance and symmetry properties for PDEs on manifolds
Full Text:

### References:

 [1] Mahdavi, A. H.; Nadjafikhah, M.; Toomanian, M., Two approaches to the calculation of approximate symmetry of ostrovsky equation with small parameter, Math Phys Anal Geom, 18, 3, 1-11 (2015) [2] Wazwaz, A. M., The tanh method for traveling wave solutions of nonlinear equations, Appl Math Comput, 154, 713-723 (2004) [3] Bluman, G. W.; Kumei, S., Symmetries and differential equations (1989), New York: Springer [4] Bluman, G. W.; Cheviakov, A. F.; Anco, S. C., Applications of symmetry methods to partial differential equations (2010), New York:Springer [5] Whitham, G. B., Linear and nonlinear waves (1974), New York: Wiley [6] Camacho, J. C.; Bruzon, M. S.; Ramirez, J.; Gandarias, M. L., Exact travelling wave solutions of a beam equation, J Nonlinear Math Phys, 18, suppl.1, 33-39 (2011) [7] Ovsiannikov, L. V., Group analysis of differential equations (1982), New York: Academic [8] Pakdemirli, M.; Yurusoy, M.; Dolapci, I. T., Comparison of approximate symmetry methods for differential equations, Acta Appl Math, 80, 3, 243-271 (2004) [9] Nadjafikhah, M.; Mokhtary, A., Approximate hamiltonian symmetry groups and recursion operators for perturbed evolution equations, Adv Math Phy, 2013 (2013) [10] Ibragimov, N. H.; Kovalev, V. F., Approximate and renormgroup symmetries, Nonlinear Physical, Science (2009), Higher Education Press, Beijing, China [11] Euler, N.; Shulga, M. W.; Steeb, W. H., Approximate symmetries and approximate solutions for amultidimensional landau-ginzburg equation, J Phys A, 25, 18 (1992) [12] Olver, P. J., Applications of lie groups to differential equations, Graduate Texts in Mathematics, 107 (1993), New York: Springer-Verlag [13] Daripa, P., Higher-order Boussinesq equations for two-way propagation of shallow water waves, Eur J Mech B/Fluids, 25, 1008-1021 (2006) [14] Daripa, P.; Hua, W., A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl Math Comput, 101, 159-207 (1999) [15] Gazizov, R. K., Lie algebras of approximate symmetries, Nonlinear Math Phys, 3, 1-2, 96-101 (1996) [16] Baikov, V. A.; Gazizov, R. K.; Ibragimov, N. H., Approximate symmetries of equations with a small parameter, Mat Sb, 136 (1988) [17] Fushchich, W. I.; Shtelen, W. M., On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter, J Phys A, 22 (1989) [18] Zakharov, V. E., On stochastization of one-dimensional chains of nonlinear oscillations, Sov Phys JETP, 38 (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.