Prolongation structures and differential forms

  • B. Kent Harrison
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 205)


Ricci Tensor Inverse Scattering Riemann Tensor Vacuum Einstein Equation Ernst Equation 
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  1. 1.
    H.D. Wahlquist and F.B. Estabrook, “Bäcklund Transformation for Solutions of the Korteweg-de Vries Equation, ”Phys. Rev. Lett. 31, 1386 (1973), and “Prolongation Structures of Nonlinear Evolution Equations”, J. Math. Phys. 16, 1 (1975).Google Scholar
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    B.K. Harrison and F.B. Estabrook, “Geometric Approach to Invariance Groups and Solution of Partial Differential Systems”, J. Math. Phys. 12, 653 (1971).Google Scholar
  3. 3.
    F.A.E. Pirani, D.C. Robinson, and W.F. Shadwick, Local Jet Bundle Formulation of Bäcklund Transformations (D. Reidel, Dordrecht, Holland, 1978). This book has an excellent list of references. See also W.F. Shadwick, “The KdV Prolongation Algebra”, J. Math. Phys. 21, 454 (1980).Google Scholar
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    B.K. Harrison, “Ernst Equation Bäcklund Transformations Revisited: New Approaches and Results,” invited paper presented at Third Marcel Grossman Meeting on the Recent Developments of General Relativity, Shanghai, China, 1982.Google Scholar
  5. 5.
    Several references on the s-G equation exist, but not too many discuss the application of the WE method to it. But see, for example, H.C. Morris, “Bäcklund Transformations and the sine-Gordon Equation”, in The 1976 Ames Research Center (NASA) Conference on the Geometric Theory of Non-Linear Waves, R. Hermann, ed. Math. Sci. Press, Brookline, Mass., 1977), p. 105.Google Scholar
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    B.K. Harrison, “Unification of Ernst-Equation Bäcklund Transformations Using a Modified Wahlquist-Estabrook Technique,” J. Math. Phys. 24, 2178 (1983).Google Scholar
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    F.B. Estabrook, “Moving Frames and Prolongation Algebras,” J. Math. Phys. 23, 2071 (1982).Google Scholar
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    P. Gragert, Symbolic Computations in Prolongation Theory, (P.K.H. Gragert, The Netherlands, 1981).Google Scholar
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    R. Hermann, “The Pseudopotentials of Estabrook and Wahlquist, The Geometry of Solitons, and the Theory of Connections,” Phys. Rev. Lett. 36, 835 (1976).Google Scholar
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    B.K. Harrison, “Bäcklund Transformation-for the Ernst Equation of General Relativity,” Phys. Rev. Lett. 41, 1197 (1978).Google Scholar
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    B.K. Harrison, “Study of the Ernst Equation Using a Bäcklund Transformation,” in Proc. Second Marcel Grossmann Meeting on General Relativity, R. Ruffini, ed. North-Holland, Amsterdam, 1982), p. 341.Google Scholar
  12. 12.
    M. Crampin, F.A.E. Pirani, and D.C. Robinson, “The Soliton Connection,” Lett. Math. Phys. 2, 303 (1978); M. Crampin, “Solitons and SL (2, R),” Phys. Lett. 66A, 170 (1978).Google Scholar
  13. 13.
    See for example H.C. Morris, “Prolongation Structure and Nonlinear Evolution Equations in Two Spatial Dimensions,” J. Math. Phys. 17, 1870 (1976), and “Inverse Scattering Problems in Higher Dimensions: Yang-Mills Fields and the Supersymmetric Sine-Gordon Equation,” J.. Math. Phys. 21, 327 (1980).Google Scholar
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    N. Sanchez, “Einstein Equations, Self-Dual Yang-Mills Fields and NonLinear Sigma Models,” Preprint, 1983.Google Scholar
  15. 15.
    B.K. Harrison, “Search for Inverse Scattering Formulation of Einstein's Vacuum Equations with one Nonnull Killing Vector,” paper presented at Ninth International Conference on General Relativity and Gravitation (GR9), Jena, DDR, 1980. *** DIRECT SUPPORT *** A3418162 00002Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • B. Kent Harrison
    • 1
  1. 1.Brigham Young UniversityProvoUSA

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