Systematic Methods for Determining the Continuous Transformation Groups Admitted by Differential Equations

  • Carl E. Wulfman


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.


Determine Equation Schroedinger Equation Contact Transformation Infinitesimal Transformation Invariance Transformation 
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  1. 1.
    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.Google Scholar
  2. 2.
    c.f. A. Cohen, An Introduction to the Lie Theory of One-parameter Groups, Stechert, NY, 1931, pp. 16–23.Google Scholar
  3. 3.
    For Lie-Backlund transformations of PDE’s with complex variables see S. Kumei, J. Math. Phys., 18, 256, (1977).CrossRefGoogle Scholar
  4. 4.
    N. Ibragimov, R. L. Anderson, J. Math. Anal. & Appl., 59, 145, (1977).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Lie, Differentialgleichungen, Leipzig, 1891, reprinted, Chelsea, NY, 1967; pp. 299–305.Google Scholar
  6. 6.
    This seems to have first been recognized by Kumei (unpub. 1974).Google Scholar
  7. 7.
    For an example see T. Shibuya, C. Wulfman, Rev. Mex. Fis. 22, 171 (1973).MathSciNetGoogle Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    c.f. J. L. Synge, Classical Physics, in Encyclopedia of Physics, S. Flugge, ed., Vol. III/1, Springer, Berlin, 1960.Google Scholar
  10. 10.
    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).MathSciNetADSGoogle Scholar
  11. 11.
    C. Wulfman, J. Phys. Al2, L73, (1979).Google Scholar
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    S. Kumei, a.) J. Math. Phys., 16, 2461, (1975); b.) ibid., 18, 256, (1977); c.) ibid. 19, 195, (1978).MathSciNetADSGoogle Scholar
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    N. H. Ibragimov, Lett. in Math. Phys. 1, 423, (1977).Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • Carl E. Wulfman
    • 1
  1. 1.Department of PhysicsUniversity of the PacificStocktonUSA

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