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Double reduction of PDEs from the association of symmetries with conservation laws with applications. (English) Zbl 1116.35004

The association of conservation laws with Noether symmetries extended to Lie-Bäcklund and nonlocal symmetries has opened the possibilities to the extension of the theory on double reductions to partial differential equations that do not have a Lagrangian and therefore to not posses Noether symmetries. at the usage of the results [A. Kara, F. Mahomed, Int. J. Theor. Phys. 39, No. 1, 23–40 (2000; Zbl 0962.35009)] the author develops the theory to effect a double reduction of PDEs with two independent variables, which is possible when the PDEs admit a symmetry associated with a conservation law. This theory is illustrated by applications to the linear heat equation, the sine-Gordon and BBM equations and a system of PDEs from one dimensional gas dynamics.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35L65 Hyperbolic conservation laws
58J70 Invariance and symmetry properties for PDEs on manifolds

Citations:

Zbl 0962.35009
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References:

[1] Kara, A.; Mahomed, F., The relationship between symmetries and conservation laws, Int. J. Theor. Phys., 39, 1, 23-40 (2000) · Zbl 0962.35009
[2] Stephani, H., Differential Equations: Their Solutions Using Symmetries (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.34001
[3] Bluman, G.; Kumei, S., Symmetries and Differential Equations. Symmetries and Differential Equations, Graduate Texts in Mathematics, vol. 81 (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0698.35001
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[5] A. Sjöberg, Non-local Symmetries and Conservation Laws for Partial Differential Equations, Thesis, University of the Witwatersrand, Gauteng, 2002.; A. Sjöberg, Non-local Symmetries and Conservation Laws for Partial Differential Equations, Thesis, University of the Witwatersrand, Gauteng, 2002.
[6] Sjöberg, A.; Mahomed, F., Non-local symmetries and conservation laws for one-dimensional gas dynamics equations, Appl. Math. Comput., 150, 2, 379-397 (2004) · Zbl 1102.76059
[7] Sjöberg, A.; Mahomed, F., The association of non-local symmetries with conservation laws: applications to the heat and Burger’s equations, Appl. Math. Comput., 168, 2, 1098-1108 (2005) · Zbl 1084.35075
[8] Steeb, W. H.; Strampp, W., Diffusion equations and Lie and Lie-Bäcklund transformation groups, Physica, 114A, 95-99 (1982) · Zbl 0513.58045
[9] A. Kara, F. Mahomed, Action of Lie-Bäcklund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997.; A. Kara, F. Mahomed, Action of Lie-Bäcklund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997.
[10] N. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1-3, Chemical Rubber Company, Boka Raton, FL, 1994-1996.; N. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1-3, Chemical Rubber Company, Boka Raton, FL, 1994-1996. · Zbl 0864.35001
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