On the classification of 2 (1 )n n   ï€dimensional non-linear Klein-Gordon equation via Lie and Noether approach
DOI:
https://doi.org/10.24297/jam.v12i10.119Keywords:
Klein-Gordon equation, Group classification, Noether approach, Conserved vectors.Abstract
A complete group classification for the Klein-Gordon equation is presented. Symmetry generators, up to equivalence transformations, are calculated for each f (u) when the principal Lie algebra extends. Further, considered equation is investigated by using Noether approach for the general case n  2. Conserved quantities are computed for each calculated Noether operator. At the end, a brief conclusion is presented.Downloads
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References
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Japan, 53 (1984) 3759-3764.
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(1981) 439-447.
2. H. Azad, M. T. Mustafa and M. Ziad, Group classification, optimal system and optimal reductions of a class of
Klein-Gordon equations, Commun. Nonlinear Sci. Numer. Simul., 15(5) (2009) 1132-1147.
3. L. F. Barannyk, A. F. Barannyk and W. I. Fushchych, Reduction of multi-dimensional Poincare invariants
nonlinear equation to two-dimenasional equations, Ukrain. Math. J., 43(10) (1991) 1311-1323.
4. G. Bluman and S. Kumei, Symmetries and Differential Equation, Springer-Verlag, NewYork, 1989.
5. A. H. Bokhari , A. Y. Al-Dweik, F. M. Mahomed and F. D. Zaman, Conservation laws of a nonlinear (n + 1)-wave
equation, Nonlinear Analysis: Real World Applications, 11(4) (2010) 2862 - 2870.
6. N. Byers, E. Noether's discovery of the deep connection between symmetries and conservation laws, Isr. Math.
Conf. Proc., 12, (1999) 67 – 82.
7. P. A. Clarkson and E. L. Mansfield, Symmetry reductions and exact solutions of a class of nonlinear heat
equation, Phys. D, 70(3) (1993) 250-288.
8. V. M. Fedorchuk, Symmetry reduction and exact solutions of a nonlinear five-dimensional wave equation, Ukrain.
Math. J., 48(4) (1996) 573-576.
9. W. I. Fushchych, On symmetry and exact solution of multi-dimensional nonlinear wave equation, Ukrain. Math. J.,
39(1) (1987) 116-123.
10. W. I. Fushchych, The symmetry of mathematical physics problems, in Algebraic-Theoretical Studies in
Mathematical Physics, Kiev, Mathematical Institute, (1981) 6-28.
11. W. I. Fushchych, Ansatz-95, J. Nonlinear Math. Phys., 2 (1995) 216-235.
12. W. I. Fushchych and I. M. Tsyfra, On a reduction and exact solutions of nonlinear wave equations with broken
symmetry, Scientific Works, 3 (2001) 251-255.
13. P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge, 2000.
14. A. Jhangeer and S. Sharif, Conservation laws for the nonlinear Klein-Gordon equation, Afr. Mat., 25(3) (2014)
833-840.
15. A. G. Johnpillai, A. H. Kara and F. M. Mahomed, Conservation laws of a nonlinear (1 + 1)-dimensional wave
equation, Nonlinear Anal. B, 11(4) (2010) 2237-2242.
16. V. Lahno, R. Zhdanov and O. Magda, Group classification and exact solutions of nonlinear wave equation, Acta
Appl. Math., 91 (2006) 253-313.
17. E. Noether, Invariante Variationsprobleme, Nacr. Konig. Gesell. Wissen., Gottingen, Math.-Phys. Kl. Heft, 2
(1918) 235 - 257. (English translation in Transport Theory and Statistical Physiscs, 1(3), 186-207).
18. E. Pucci, Group analysis of the equation ( , ) tt xx x u ï¬u  g u u , Riv. Mat. Univ. Parma, 4 (1987) 71-87.
19. P. Rudra, Symmetry group of the nonlinear Klein-Gordon equation, J. Phys. A: Math. Gen., 19(13) (1986) 2499-
2504.
20. M. Tajiri, Some remarks on similarity and soliton solutions of nonlinear Klein-Gordon equations, J.Phys. Soc.
Japan, 53 (1984) 3759-3764.
21. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
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Published
2016-11-30
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Jhangeer, A., & Al-Mufadi, F. (2016). On the classification of 2 (1 )n n   ï€dimensional non-linear Klein-Gordon equation via Lie and Noether approach. JOURNAL OF ADVANCES IN MATHEMATICS, 12(10), 6720–6727. https://doi.org/10.24297/jam.v12i10.119
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