IJPAM: Volume 108, No. 2 (2016)

OPTIMAL SYSTEM AND SYMMETRY REDUCTION OF
THE $(1+1)$ DIMENSIONAL SAWADA-KOTERA EQUATION

N. Kadkhoda$^1$, H. Jafari$^{2,3}$, G.M. Moremedi$^3$, D. Baleanu$^{4,5}$
$^1$Department of Mathematics
Faculty of Basic Sciences
Bozorgmehr University Of Qaenat
Qaenat, IRAN
$^2$Department of Mathematics
University of Mazandaran
Babolsar, IRAN
$^3$Department of Mathematical Sciences
University of South Africa
UNISA0003, Pretoria, SOUTH AFRICA
$^{4}$Department of Mathematics and Computer Sciences
Faculty of Art and Sciences
Cankaya University, Ankara, TURKEY
$^5$Institute of Space Sciences
76900 Magurele-Bucharest, ROMANIA

Abstract. We study the nonlinear fifth order $(1+1)$ dimensional Sawada-Kotera equation using Lie symmetry group. For this equation Lie point symmetry operators and optimal system are obtained. We determine the corresponding invariant solutions and reduced equations using obtained infinitesimal generators.

Received: September 4, 2015

Revised: September 4, 2015

Published: October 1, 2016

AMS Subject Classification: 70G65

Key Words and Phrases: Lie symmetry, optimal system, Sawada-Kotera equation, group-invariant solutions
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Bibliography

1
E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218.

2
A.M. Wazwaz, The tanh-coth method for solitons and kink solutions fornon-linear parabolic equations, Appl. Math. and Comp., 188 (2007), 1467-1475.

3
H. Jafari, N. Kadkhoda, A. Biswas, The $( \frac{G'}{G})$-expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres, Jour. of King Saud Uni-Sci, (2013) 25, 57-62.

4
E. Salehpour, H. Jafari, N. Kadkhoda, Application of $( \frac{G'}{G})$-expansion method to nonlinear Lienard equation, Indian Journal of Sci. and Tech., 5 (2012), 2554-2556.

5
H. Jafari, A. Borhanifar, S.A. Karimi, New solitary wave solutions for the bad Boussinesq and good Boussinesq equations, Numer. Meth. for Par. Diff. Eq., 25, No. 5 (2009), 1231-1237.

6
A.M. Wazwaz, The sine-cosine method for obtaining solutions with compact and noncompact structures, Appl. Math. and Comp., 159 (2004), 559-576.

7
H. Jafari, N. Kadkhoda, C.M. Khalique, Travelling wave solutions of nonlinear evolution equations using the simplest equation method, Comp. and Math. with Appl., 64, No. 6 (2012), 2084-2088.

8
K. Sawada, T. Kotera, Prog. Theor. Phys., 51 (1974), 1355.

9
B. Fuchssteiner, W. Oevel, J. Math. Phys., 23 (1982), 358.

10
R.N. Aiyer, B. Fuchssteiner, W. Oevel, J. Phys. A, 19 (1986), 3755.

11
D. Levi, O. Ragnisco, Inverse Problems, 4 (1988), 815.

12
C. Liu, Z. Dai, Exact soliton solutions for the fifth-order Sawada-Kotera equation, Appl. Math. and Comp., 206 (2008), 72-275.

13
Q. Feng, B. Zheng, Traveling Wave Solutions for the Fifth-Order Sawada-Kotera Equation and the General Gardner Equation by $( \frac{G'}{G})$-Expansion Method, WSEAS Tran. on Math., 9, No. 3 (2010), 171-180.

14
A.M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. and Comp., 184 (2007), 1002-1014.

15
N.H. Ibragimov, Handbook of Lie Group Analysis of Differential Equations, Volumes 1, 2, 3, CRC Press, Boca Raton, Ann Arbor, London, Tokyo (1994,1995,1996).

16
L.V. Ovsyannikov, Group Analysis of Differential Equations, Academic Pres, New York (1982).

17
P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, 2-nd Edition, Springer-Verlag, Berlin (1993).

18
G. Baumann, Symmetry Analysis of Differential Equations with Mathematica, Telos, Springer Verlag, New York (2000).

19
G.W. Bluman, S.C. Anco, Symmetry and Integtation Methods for Differential Equations, Springer-Verlag, New York (2002).

20
M. Nadjafikhah, F. Ahangari, Symmetry Reduction of Two-Dimensional Damped Kuramoto-Sivashinsky Equation, Commun. Theor. Phys., 56 (2011), 211-217.

21
P.E. Hydon, Symmetry Methods For Differential Equations A Beginner's Guide, Cambridge (2000).

22
G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81, Springer-Verlag, New York (1989).

23
G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990).

24
M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, In: Operator Theory: Advances and Applications, Birkhäuser, Basel (1994), 369-396.

25
D.S. Moak, The $q$-analogue of the Laguerre polynomials, J. Math. Anal. Appl., 81 (1981), 20-47.

26
A.G. Johnpillai, C.M. Khalique, Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1207-1215.

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DOI: 10.12732/ijpam.v108i2.3 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 108
Issue: 2
Pages: 215 - 226


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