Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 001, 12 pages      math-ph/0510068

The Differential Form Method for Finding Symmetries

B. Kent Harrison
Physics and Astronomy Department, Brigham Young University, Provo, Utah 84602, USA

Received July 20, 2005; Published online August 03, 2005

This article reviews the use of differential forms and Lie derivatives to find symmetries of differential equations, as originally presented in Harrison and Estabrook, J. Math. Phys., 12 (1971), 653. An outline of the method is given, followed by examples and references to recent papers using the method.

Key words: symmetries; differential equations; differential forms.

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  1. Harrison B.K., Estabrook F.B., Geometric approach to invariance groups and solution of partial differential systems, J. Math. Phys., 1971, V.12, 653-666 (Paper I).
  2. Harrison B.K., Differential form symmetry analysis of two equations cited by Fushchych, in Proceedings of Second International Conference "Symmetry in Nonlinear Mathematical Physics" (July 7-13, 1997, Kyiv), Editors M.I. Shkil, A.G. Nikitin and V.M. Boyko, V.1, 21-33 (Paper II).
  3. Edelen D.G.B., Programs for calculation of isovector fields in the REDUCE.2 environment, Center for the Application of Mathematics, Lehigh University, 1981.
  4. Gragert P.K.H., Symbolic computations in prolongation theory, Ph.D. Thesis, Twente University of Technology, Enschede, The Netherlands, 1981.
  5. Kersten P.H.M., Gragert P.K.H., Symbolic integration of overdetermined systems of partial differential equations, Memorandum 430, Twente University of Technology, Enschede, The Netherlands, 1983.
  6. Gragert P.K.H., Kersten P.H.M., Martini A., Symbolic computations in applied differential geometry, Acta Appl. Math., 1983, V.1, 43-77.
  7. Kersten P.H.M., Infinitesimal symmetries: a computational approach, Ph.D. Thesis, Twente University of Technology, Enschede, The Netherlands, 1985.
  8. Kersten P.H.M., Software to compute infinitesimal symmetries of exterior differential systems, with applications, Acta Appl. Math., 1989, V.16, 207-229.
  9. Gragert P.K.H., Kersten P.H.M., Differential geometric computations and computer algebra, Math. Comp. Modelling, 1997, V.25, 11-24.
  10. Kersten P.H.M., Gragert P.K.H., The Lie algebra of infinitesimal symmetries of nonlinear diffusion equation, J. Phys. A: Math. Gen., 1983, V.16, L685-L688.
  11. Kersten P.H.M., Martini R., Lie-Bäcklund transformations for the massive Thirring model, J. Math. Phys., 1985, V.26, 822-825.
  12. Kersten P.H.M., Creating and annihilating Lie-Bäcklund transformations of the Federbush model, J. Math. Phys., 1986, V.27, 1139-1144.
  13. Langton B.T., Lie symmetry techniques for exact interior solutions of the Einstein field equations for axially symmetric, stationary, rigidly rotating perfect fluids, Ph.D. Thesis, University of Sydney, Australia, 1997.
  14. Carminati J., Devitt J.S., Fee G.J., Isogroups of differential-equations using algebraic computing, J. Symbolic Comp., 1992, V.14, 103-120.
  15. Heck A., Introduction to Maple, Springer, New York, 1993, 413-417.
  16. Char B.W., Geddes K.O., Gonnet G.H., Leong B.L., Monagan M.B., Watt S.M., Maple V library reference manual, New York, Springer, 1991, 540-560.
  17. Neilsen D.W., A search for an interior solution in general relativity using Lie-Bäcklund symmetries, M.S. Thesis, Provo (Utah, USA), Brigham Young University, 1995.
  18. Hereman W., Review of symbolic software for Lie symmetry analysis, Math. Comp. Modelling, 1997, V.25, 115-132.
  19. Steeb W.H., Symmetries and vacuum Maxwell's equations, J. Math. Phys., 1980, V.21, 1656-1680.
  20. Steeb W.H., Erig W., Strampp W., Symmetries and the Dirac equation, J. Math. Phys., 1981, V.22, 970-973.
  21. Satir A., Duff-Inami-Pope-Sezgin-Stelle bosonic membrane equations as an involutory system, Progr. Theoret. Phys., 1998, V.100, 1273-1280.
  22. Bluman G.W., Cole J.D., The general similarity solution of the heat equation, J. Math. Mech., 1969, V.18, 1025-1042.
  23. Webb G.M., Lie symmetries of a coupled nonlinear Burgers heat-equation system, J. Phys. A: Math. Gen., 1990, V.23, 3885-3894.
  24. Stephani H., Differential equations. Their solution using symmetries, Cambridge, Cambridge University Press, 1989, 28.
  25. Papachristou C.J., Harrison B.K., Isogroups of differential ideals of vector-valued differential forms: application to partial differential equations, Acta Appl. Math., 1988, V.11, 155-175.
  26. Papachristou C.J., Harrison B.K., Symmetry groups of partial differential equations associated with vector-valued differential forms, in Proceedings of the XV International Colloquium in Group Theoretical Methods in Physics, Editor R. Gilmore, Singapore, World Scientific, 1987, 440-445.
  27. Papachristou C.J., Harrison B.K., Some aspects of the isogroup of the self-dual Yang-Mills system, J. Math. Phys., 1987, V.28, 1261-1264.
  28. Papachristou C.J., Harrison B.K., Nonlocal symmetries and Bäcklund transformations for the self-dual Yang-Mills system, J. Math. Phys., 1988, V.29, 238-243.
  29. Waller S.M., A three-parameter group similarity solution for a one dimensional nonlinear diffusion equation, Phys. Scripta, 1990, V.41, 193-196.
  30. Waller S.M., Isogroup and general similarity solution of a nonlinear diffusion equation, J. Phys. A: Math. Gen., 1990, V.23, 1035-1040.
  31. Waller S.M., Invariant group similarity solution for a class of reaction-diffusion-equations, Phys. Scripta, 1990, V.42, 385-388.
  32. Edelen D.G.B., Isovector methods for equations of balance. With programs for computer assistance in operator calculations and an exposition of practical topics of the exterior calculus, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, The Hague, Martinus Nijhoff Publishers, 1980.
  33. Edelen D.G.B., Applied exterior calculus, New York, John Wiley and Sons, 1985, Chap. 6.
  34. Edelen D.G.B., Isovector fields for problems in the mechanics of solids and fluids, Internat. J. Engrg. Sci., 1982, V.20, 803-815.
  35. Edelen D.G.B., On solving problems in the mechanics of solids and fluids by a generalized method of characteristics, Internat. J. Engrg. Sci., 1988, V.26, 361-372.
  36. Edelen D.G.B., Order-independent method of characteristics, Internat. J. Theoret. Phys., 1989, V.28, 303-333.
  37. Edelen D.G.B., Implicit similarities and inverse isovector methods, Arch. Rat. Mech. and Anal., 1983, V.82, 181-189.
  38. Webb G.M., Brio M., Zank G.P., Symmetries of the triple degenerate DNLS equations for weakly nonlinear dispersive MHD waves, J. Plasma Phys., 1995, V.54, 201-244.
  39. Webb G.M., Similarity considerations and conservation laws for magneto-static atmospheres, Solar Phys., 1986, V.106, 287-313.
  40. Pakdemirli M., Yürüsoy M., Küçükbursa A., Symmetry groups of boundary layer equations of a class of non-Newtonian fluids, Internat. J. Non-Linear Mech., 1996, V.31, 267-276.
  41. Pakdemirli M., Yürüsoy M., Equivalence transformations applied to exterior calculus approach for finding symmetries: an example of non-Newtonian fluid flow, Internat. J. Engrg. Sci., 1999, V.37, 25-32.
  42. Suhubi E.S., Chowdhury K.L., Isovectors and similarity solutions for nonlinear reaction-diffusion equations, Internat. J. Engrg. Sci., 1988, V.26, 1027-1041.
  43. Suhubi E.S., Bakkaloglu A., Group properties and similarity solutions for a quasi-linear wave-equation in the plane, Internat. J. Non-Linear Mech., 1991, V.26, 567-584.
  44. Suhubi E.S., Isovector fields and similarity solutions for general balance-equations, Internat. J. Engrg. Sci., 1991, V.29, 133-150.
  45. Pakdemirli M., Suhubi E.S., Similarity solutions of boundary-layer equations for second-order fluids, Internat. J. Engrg. Sci., 1992, V. 30, 611-629.
  46. Suhubi E.S., Equivalence groups for second order balance equations, Internat. J. Engrg. Sci., 1999, V.37, 1901-1925.
  47. Ozer S., Suhubi E.S., Equivalence transformations for first order balance equations, Internat. J. Engrg. Sci., 2004, V.42, 1305-1324.
  48. Suhubi E.S., Equivalence groups for balance equations of arbitrary order, Part I, Internat. J. Engrg. Sci., 2004, V.42, 1729-1751.
  49. Bhutani O.P., Bhattacharya L., Isogroups and exact-solutions for some Klein-Gordon and Liouville-type equations in n-dimensional Euclidean space, J. Math. Phys., 1995, V.36, 3759-3770.
  50. Bhutani O.P., Vijayakumar K., On the isogroups of the generalized diffusion equation, Internat. J. Engrg. Sci., 1990, V.28, 375-387.
  51. Bhutani O.P., Singh K., On certain exact solutions of a generalized K-dV-Burger type equation via isovector method-I, Internat. J. Engrg. Sci., 2000, V.38, 1741-1753.
  52. Chowdhury K.L., On the isovectors of a class of nonlinear diffusion-equations, Internat. J. Engrg. Sci., 1986, V.24, 1597-1605.
  53. Chowdhury K.L., A note on the isovector of rate-type materials, Internat. J. Engrg. Sci., 1986, V.24, 819-826.
  54. Delph T.J., Isovector fields and self-similar solutions for power law creep, Internat. J. Engrg. Sci., 1983, V.21, 1061-1067.
  55. Vijayakumar K., Isogroup classification of equations of meteorology, Internat. J. Engrg. Sci., 1996, V.34, 1157-1164.
  56. Hu Z-J., On the isovectors of the principal chiral model, J. Math. Phys., 1991, V.32, 2540-2542.
  57. Barco M.A., An application of solvable structures to classical and nonclassical similarity solutions, J. Math. Phys., 2001, V.42, 3714-3734.
  58. Kalpakides V.K., Isovector fields and similarity solutions of nonlinear thermoelasticity, Internat. J. Engrg. Sci., 1998, V.36, 1103-1126.
  59. Harnad J., Winternitz P., Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation, J. Math. Phys., 1982, V.23, 517-525.

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