Approximate symmetry and solutions of the nonlinear Klein-Gordon equation with a small parameter. (English) Zbl 1367.35016
Summary: In this paper, the Lie approximate symmetry analysis is applied to investigate new solutions of the nonlinear Klein-Gordon equation with a small parameter. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with a 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 |
Keywords:
exact solutions; approximate solutions; hyperbolic function method; Riccati equation methodReferences:
| [1] | V. A. Baikov, R. K. Gazizov and N. H. Ibragimov, Approximate symmetries of equations with a small parameter, Mat. Sb.136 (1988) 435-450; (in Russian) Math. USSR Sb.64 (1989) 15. · Zbl 0675.35018 |
| [2] | Gazizov, R. K., Lie algebras of approximate symmetries, Nonlinear Math. Phys.3(12) (1996) 96-101. · Zbl 1044.58507 |
| [3] | Fushchich, W. I. and Shtelen, W. M., On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter, J. Phys. A22 (1989) 887-890. · Zbl 0711.35086 |
| [4] | Euler, N., Shulga, M. W. and Steeb, W. H., Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation, J. Phys. A25(18) (1992) 1095-1103. · Zbl 0764.35096 |
| [5] | Pakdemirli, M., Yurusoy, M. and Dolapci, I. T., Comparison of approximate symmetry methods for differential equations, Acta Appl. Math.80(3) (2004) 243-271. · Zbl 1066.76052 |
| [6] | Nadjafikhah, M. and Mokhtary, A., Approximate Hamiltonian symmetry groups and recursion operators for perturbed evolution equations, Adv. Math. Phys.2013 (2013) 9 pp. · Zbl 1350.35073 |
| [7] | Camacho, J. C., Bruzon, M. S., Ramirez, J. and Gandarias, M. l., Exact travelling wave solutions of a beam equation, J. Nonlinear Math. Phys.18(Suppl. 1) (2011) 33-49. · Zbl 1362.35298 |
| [8] | Wazwaz, A. M., The tanh method for traveling wave solutions of nonlinear equations, Appl. Math. Comput.154 (2004) 713-723. · Zbl 1054.65106 |
| [9] | Wazwaz, A. M., The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comput.188 (2007) 1467-1475. · Zbl 1119.65100 |
| [10] | Wazwaz, A. M., New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Commun. Nonlinear Sci. Numer. Simul.13 (2008) 889-901. · Zbl 1221.35372 |
| [11] | Luke, J. C., Perturbation method for nonlinear dispersive wave problems, Proc. Roy. Soc. London292 (1966) 403-412. · Zbl 0143.13603 |
| [12] | Parkes, E. J. and Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun.98 (1996) 288-300. · Zbl 0948.76595 |
| [13] | Malfliet, W., Solitary wave solutions of nonlinear wave equations, Amer. J. Phys.60(7) (1992) 650-654. · Zbl 1219.35246 |
| [14] | Malfliet, W., The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scripta54 (1996) 563-568. · Zbl 0942.35034 |
| [15] | Olver, P. J., Applications of Lie Groups to Differential Equations, , Vol. 107 (Springer-Verlag, New York, 1993). · Zbl 0785.58003 |
| [16] | Bluman, G. W. and Kumei, S., Symmetries and Differential Equations (Springer, New York, 1989). · Zbl 0698.35001 |
| [17] | Bluman, G. W., Cheviakov, A. F. and Anco, S. C., Applications of Symmetry Methods to Partial Differential Equations (Springer-Verlag, New York, 2010). · Zbl 1223.35001 |
| [18] | Ibragimov, N. H. and Kovalev, V. F., Approximate and Renormgroup Symmetries, (Higher Education Press, Beijing, China, 2009). · Zbl 1170.22001 |
| [19] | Ostrovsky, L., Asymptotic Perturbation Theory of Waves (Imperial College Press, London, 2015), pp. 61-68. · Zbl 1320.35003 |
| [20] | Ovsiannikov, L. V., Group Analysis of Differential Equations (Academic Press, New York, 1982). · Zbl 0485.58002 |
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