Abstract
This talk will present some recent results obtained by using Lie’s systematic methods to uncover transformation groups admitted by several types of differential equations. We begin by sketching the methods.
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References
The variables zr take on hyperreal values as well, since zr + dzr is also a zr: c.f.e.g., K. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, NY, 1976.
c.f. A. Cohen, An Introduction to the Lie Theory of One-parameter Groups, Stechert, NY, 1931, pp. 16–23.
For Lie-Backlund transformations of PDE’s with complex variables see S. Kumei, J. Math. Phys., 18, 256, (1977).
N. Ibragimov, R. L. Anderson, J. Math. Anal. & Appl., 59, 145, (1977).
S. Lie, Differentialgleichungen, Leipzig, 1891, reprinted, Chelsea, NY, 1967; pp. 299–305.
This seems to have first been recognized by Kumei (unpub. 1974).
For an example see T. Shibuya, C. Wulfman, Rev. Mex. Fis. 22, 171 (1973).
C. Wulfman, T. Sumi in Atomic Scattering Theory, J. Nuttall, ed., U. of Western Ontario, London, Ont., 1978; pp. 197202. See Also C. Wulfman, Dynamical Groups in Atomic and Molecular Physics, in Recent Advances in Group Theory and Their Application to Spectroscopy, J. Donini, ed., Plenum, NY, 1979.
c.f. J. L. Synge, Classical Physics, in Encyclopedia of Physics, S. Flugge, ed., Vol. III/1, Springer, Berlin, 1960.
R. L. Anderson, S. Kumei, C. Wulfman; a.) Phys. Rev. Lett., 28, 988, 1972; b.) Rev. Mex. Fix. 21, 1, (1972); c.) Rev. Mex. Fis. 21, 35, (1972); d.) J. Math. Phys. 14, 1527 (1973).
C. Wulfman, J. Phys. Al2, L73, (1979).
S. Kumei, a.) J. Math. Phys., 16, 2461, (1975); b.) ibid., 18, 256, (1977); c.) ibid. 19, 195, (1978).
N. H. Ibragimov, Lett. in Math. Phys. 1, 423, (1977).
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© 1980 Plenum Press, New York
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Wulfman, C.E. (1980). Systematic Methods for Determining the Continuous Transformation Groups Admitted by Differential Equations. In: Gruber, B., Millman, R.S. (eds) Symmetries in Science. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-3833-8_28
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DOI: https://doi.org/10.1007/978-1-4684-3833-8_28
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