Research ArticleOpen AccessGoogle Scholar indexed
A Note on the Lie Algebra of the Invariants in the CBS Nonlinear Equation
Fsica Teórica, Facultad de Ciencias, Universidad de Salamanca, Salamanca, Spain
- 1 Fsica Teórica, Facultad de Ciencias, Universidad de Salamanca, Salamanca, Spain
Journal of Modern Physics·Volume 09 (2018)·Pages 1297–1303·Published 9 May 2018·DOI10.4236/jmp.2018.96078
Copy link · social · email
Abstract
In this short note, a particular realization of the vector fields that form a Lie Algebra of symmetries for the Calogero-Bogoyavleskii-Schiff equation is found. The Lie Algebra is examined and the result is a semidirect product of two Lie Groups. The structure of the semidirect product is examined through the table of commutation rules. Two reductions are made with the help of two sets of generators and the final outcome for the solution is related to the elliptic Painlevé P( ξ )-function.
KeywordsNonlinear IntegrabilityGroup TheoryWeierstrass P-Elliptic Function
- Bogoyavlenskii, O. (1990) Russian Mathematical Surveys, 45, 1-86.
- Peng, Y. (2006) International Journal of Theoretical Physics, 45, 1779-1783. https://doi.org/10.1007/s10773-006-9139-7
- Saleh, R., Kassem, M. and Mabrouk, S. (2017) Mathematical Methods in the Applied Sciences, 40, 5851-5862. https://doi.org/10.1002/mma.4435
- Wazwaz, A. (2008) Applied Mathematics and Computation, 196, 363-370. (See also 203, 592-597)
- Kumar, R. (2016) IOSR Journal of Mathematics, 12, 144-147. https://doi.org/10.9790/5728-120402144147
- Olver, P.J. (1986) Applications of Lie Groups to Differential Equations. 2nd Edition, Springer Verlag, Berlin.
- Weiss, J., Tabor, M. and Carnevale, G. (1983) Journal of Mathematical Physics, 24, 522-536. https://doi.org/10.1063/1.525721
- Weiss, J. (1983) Journal of Mathematical Physics, 24, 1405-1423. https://doi.org/10.1063/1.525875
- Estévez, P.G., Prada, J. and Villarroel, J. (2007) Journal of Physics A: Mathematical and Theoretical, 40, 7213-7231. https://doi.org/10.1088/1751-8113/40/26/008
- Hammermesh, M. (1964) Group Theory and its Applications to Physics Problems. 2nd Edition, Addison-Wesley Publishing Co., Inc., Boston.
- Cerveró, J.M. and Polo, P.P. (2016) European Journal of Physics, 37, 055401.
- Dilatations and the Conformal Group Are Treated in the Recent Report by H. Dreyer. http://edu.itp.phys.ethz.ch/fs13/cft/CGIVD2_Dreyer.pdf
- Byrd, P.F. and Friedman, M.D. (1971) Handbook of Elliptic Functions for Engineers and Scientists. 2nd Edition, Springer Verlag, Berlin. https://doi.org/10.1007/978-3-642-65138-0
- Clarkson, P.A. (2006) The Painlevè Equations—Nonlinear Special Functions. In: Marcellán, F. and van Assche, W., Eds., Orthogonal Polynomials and Special Functions: Computation and Applications? Lecture Notes in Mathematics, 1883. Springer Verlag, Berlin, 331-411.