Abstract
Classification of optimal group-invariant solutions has been carried out for cylindrical Korteweg–de Vries equation. It is a nonlinear evolution equation often found in studies of fluid mechanics and plasmas. An optimal system of subalgebras is obtained for the group of Lie generators associated with the equation, and then, the group-invariant solutions are obtained for each member in the optimal class. The procedure for calculating other group-invariant solutions from the optimal class is also illustrated with an example.
References
Maxon, S., Viecelli, J.: Cylindrical solitons. Phys. Fluids 17, 1614–1616 (1974)
Mamun, A.A., Shukla, P.K.: Cylindrical and spherical dust ion-acoustic solitary waves. Phys. Plasmas 9, 1468–1470 (2001)
Meneses, H.P., Chwang, A.T.: IIHR report no 247, University of Iowa (1982)
Calogero, F., Degasperis, A.: Spectral Transforms and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations-I. North-Holland Publishing Company, Amsterdam (1982)
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)
Olver, P.J.: Applications of Lie groups to Differential Equations. Springer, New York (1993)
Ibragimov, N.H.: Lie Group Analysis of Differential Equations, vol. 1. CRC Press, Boca Raton (1994)
Qian, M., Hu, X.R., Chen, Y.: ONEOptimal: a Maple package for generating one-dimensional optimal system of finite dimensional Lie algebra. Commun. Theor. Phys. 61(2), 160 (2014)
Hu, X.R., Li, Y.Q., Chen, Y.: A direct algorithm of one-dimensional optimal system for the group invariant solutions. J. Math. Phys. 56(5), 053504 (2015)
Hu, X.R., Li, Y.Q., Chen, Y.: Constructing two-dimensional optimal system for the group invariant solutions. J. Math. Phys. 57(2), 023518 (2016)
Zakharov, N.S., Korobeinikov, V.P.: Group analysis of the generalised Korteweg–de Vries–Burger’s equation. J. Appl. Math. Mech. 44, 668 (1980)
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Communicated by Francesco Zirilli.
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Gupta, S.K., Ghosh, S.K. Classification of Optimal Group-Invariant Solutions: Cylindrical Korteweg–de Vries Equation. J Optim Theory Appl 173, 763–769 (2017). https://doi.org/10.1007/s10957-017-1111-6
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DOI: https://doi.org/10.1007/s10957-017-1111-6
Keywords
- Cylindrical Korteweg–de Vries equation (cKdV)
- One-parameter Lie generators
- Adjoint generators
- Optimal system of subalgebra
- Optimal group-invariant solutions