Abstract
New results on hidden and nonlocal symmetries of nonlinear ordinary differential equations (NLODEs) are presented. Two types of hidden symmetries have been identified. A type I (II) hidden symmetry of an ODE occurs if a symmetry is lost (gained) when the order of the ODE is reduced. Both type I and type II hidden symmetries are found in the reduction of a third-order NLODE invariant under a three-parameter nonsolvable Lie group. Nonlocal group generators are determined of the exponential form and a new linear form. The ODEs can be reduced by the nonlocal group generators until first-order ODEs are obtained where the procedure fails because canonical coordinates cannot be calculated in that case. ODEs cannot be reduced by the linear nonlocal group generators.
Supported in part by a grant from the Southwestern Bell Corporation.
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References
R. L. Anderson and N. H. Ibragimov, Lie-Bäcklund Transformations in Applications, SIAM, Philadelphia, 1979.
G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Appl. Math. Sci. No. 81, Springer-Verlag, New York, 1989.
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
G. W. Bluman and J. D. Cole, Similarity Methods for differential Equations, Springer-Verlag, 1974.
P. J. Olver and P. Rosenau, SIAM J. Appl. Math. 47, 263–275, 1987.
P. Clarkson and M. Kruskal, “New similarity reductions of the Boussinesq equation,” J. Math. Phys. 30, 2201–2213, 1989.
A. Cohen, An Introduction to the Lie theory of One-Parameter Groups with Applications to the Solution of differential Equations, D. C. Heath, New York, 1911.
B. Abraham-Shrauner and Ann Guo, “Hidden Symmetries Associated with the Projective Group of Nonlinear First-Order Ordinary Differential Equations,” J. Phys. A. 25, 5597, 1992.
A. Guo and B. Abraham-Shrauner, “Hidden Symmetries of Energy Conserving Differential Equations,” IMA J. Appl. Math. (submitted for publication).
B. Abraham-Shrauner and Ann Guo, “Hidden symmetries of Differential Equations,” Proceedings of the AMS March, 1992 meeting in Springfield, Missouri (submitted).
B. Abraham-Shrauner and P. G. L. Leach, “Hidden Symmetries of Nonlinear Ordinary Differential Equations,” AMS-SIAM Summer Seminar Proceedings (submitted).
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© 1993 Springer Science+Business Media Dordrecht
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Abraham-Shrauner, B., Guo, A. (1993). Hidden and Nonlocal Symmetries of Nonlinear Differential Equations. In: Ibragimov, N.H., Torrisi, M., Valenti, A. (eds) Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2050-0_1
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DOI: https://doi.org/10.1007/978-94-011-2050-0_1
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