On the Behaviour of Solutions of the Laplace-Beltrami Equations*

  • A. M. Grundland**


This paper presents an approach to the nonlinar second order P.D.E.’s based on group theoretical and differential geometric methods - namely symmetry reduction method and Riemann invariants method which provide us with complementary results. The presentation here is based on the results of the previous papers1,2 (with some modifications) and is their continuation.


Minkowski Space Isotropic Solution Isotropic Vector Symmetry Variable Poincare Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. M. Grundland, J. Harnad, P. Winternitz, Symmetry reduction for nonlinear relativistically invariant equations, J. Math. Phys. 25, 4, 791–806, (1984).CrossRefGoogle Scholar
  2. 2.
    G. Cieciura, A. M. Grundland, A certain class of solutions of the nonlinear wave equation, J. Math. Phys. 25, 12, 3460–3469, (1984).CrossRefGoogle Scholar
  3. 3.
    A. M. Grundland, J. Hamad, P. Winternitz, Solutions of the multidimensional Sine Gordon equation obtained by symmetry reduction, KINAM Rev. Fis. 4, 333–344, (1982).Google Scholar
  4. 4.
    A. M. Grundland, J. A. Tuszyriski, P. Winterritz, Solutions of the multidimensional classical 06 field equations, (to be published).Google Scholar
  5. 5.
    Z. Peradzynski, Riemann invariants for the nonplanar k-waves, Bull. Acad. Pol. Sci. 29, 10, 67–74, (1971).Google Scholar
  6. 6.
    P. J. Olver, Symmetry groups and group invariant solutions of partial differential equations, J. Diff. Geom. 14, 497–542, (1979).Google Scholar
  7. 7.
    C. B. Collins, All solutions to a nonlinear system of complex potential equations, J. Math. Phys. 21, 2, 240–248, (1980).CrossRefGoogle Scholar
  8. 8.
    C. B. Collins, Complex potential equations, special relativity and complexified Minkowski space-time, J. Math. Phys. 21, 2, 249–255, (1980).CrossRefGoogle Scholar
  9. 9.
    F. G. Friedlander, The wave equation on a curved space, Cambridge Univ. Press, 1975.Google Scholar
  10. 10.
    S. Helgason, A formula for the radial part of the Laplace-Beltrami operator, J. Diff. Geom. 6, 411–419, (1972).Google Scholar
  11. 11.
    S. Helgason, Wave equations on homogeneous space, Acta Math. 9, 1–33, (1984).Google Scholar
  12. 12.
    H. Urbantke, On complex-valued scalar waves of the simply-progressive type, Report from the Institute of Theoretical Physics, Vienna University, 15, 1–11, (1981).Google Scholar
  13. 13.
    G. W. Bluman, J. D. Cole, Similarity methods for differential equations, Springer-Verlag, New York, 1974.CrossRefGoogle Scholar
  14. 14.
    L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York, 1982.Google Scholar
  15. 15.
    N. H. Ibragimov, Transformation groups applied to mathematical physics, D. Reidel, Boston, 1984.Google Scholar
  16. 16.
    P. Winternitz, Lie groups and solutions of nonlinear differential equations in Nonlinear Phenomena, Lecture notes in Physics, Springer-Verlag, New York, 263, 1982.Google Scholar
  17. 17.
    P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math. 25, 1, 1–85, (1902).CrossRefGoogle Scholar
  18. 18.
    B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes, Acta Math. 33, 1, 1–55, (1910).CrossRefGoogle Scholar
  19. 19.
    M. J. Ablowitz, A. Ramani, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, J. Math. Phys. 21, 4, 715–721 and 1006–1021, (1980).Google Scholar
  20. 20.
    V. Golubev, Lectures on integrals of equations of motion, State Publ. House, Moscow, 1953.Google Scholar
  21. 21.
    H. Davis, Introduction to nonlinear differential and integral equations, Dover, New York, 1962.Google Scholar
  22. 22.
    J. Patera, P. Winternitz, R. Sharp, H. Zassenhaus, Subgroups of the similitude group of three-dimensional Minkowski space, Can. J. Phys., 54, 9, 950–961, (1976).CrossRefGoogle Scholar
  23. 23.
    J. Patera, P. Winternitz, H. Zassenhaus, Continuous subgroups of the fundamental groups of physics II. The similitude groups, J. Math. Phys. 16, 8, 1615–1623, (1975).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • A. M. Grundland**
    • 1
  1. 1.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada

Personalised recommendations