Skip to main content
SpringerLink
Log in

Lie Groups and Solutions of Nonlinear Partial Differential Equations

  • Chapter
Integrable Systems, Quantum Groups, and Quantum Field Theories

Part of the book series: NATO ASI Series ((ASIC,volume 409))

Abstract

The application of local Lie point transformation groups to the solution of partial differential equations is reviewed. The method of symmetry reduction is presented as an algorithm. Included is the construction of group invariant solutions, partially invariant solutions and also the use of conditional symmetries. The emphasis in on recent developments, including the use of computer algebra. Many examples and applications are treated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Olver P.J., Applications of Lie Groups to Differential Equations, ( Springer, New York, 1986 ).

    Book  MATH  Google Scholar 

  2. Bluman G.W. and Cole J.D., Similarity Methods for Differential Equations, ( Springer, New York. 1974 ).

    Book  MATH  Google Scholar 

  3. Bluman G.W. and Kumei S.,Symmetries and Differential Equations, ( Springer, Berlin 1989 ).

    MATH  Google Scholar 

  4. Ovsiannikov, L.V., Group Analysis of DifferentialEquation, ( Academic, New York, 1982 ).

    Google Scholar 

  5. Ibragimov N.H., Transformation Groups Applied toMathematical Physics, ( Reidel, Boston, 1985 ).

    Book  MATH  Google Scholar 

  6. Anderson R.L. and Ibragimov N.H., Lie-Backlund Transformations in Applications, ( SIAM, Philadelphia, 1979 ).

    Book  Google Scholar 

  7. Ames W.F., Nonlinear Partial Differential Equations in Engineering, ( Academic, New York, 1972 ).

    MATH  Google Scholar 

  8. Winternitz P., “Group Theory and Exact Solutions of Partially Integrable Differential Systems”, in R. Conte and N. Boccara (Editors) Partially Integrable Evolution Equations in Physics, ( Kluwer Academic Publishers, Netherlands, 1990 ).

    Google Scholar 

  9. Levi, D. and Winternitz, P. (Editors), Symmetries and Nonlinear Phenomena, ( World Scientific, Singapore, 1988 ).

    MATH  Google Scholar 

  10. Ablowitz M.J., and Segur H., Solitons and the InverseScattering Transform, (SIAM, Philadelphia, 1981 ).

    Book  MATH  Google Scholar 

  11. Ablowitz M.J., and Clarkson P.A., Solitons, Nonlinear Evolution Equations andInverse Scattering, ( Cambridge University Press, Cambridge, 1991 ).

    Book  MATH  Google Scholar 

  12. Schwarz F., Computing 34, 91 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  13. Champagne B. and Winternitz P., Preprint CRM-1278, Montrai, 1985.

    Google Scholar 

  14. Champagne B., Hereman W. and Winternitz P., Comp. Phys. Commun. 66, 319 (1991).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Brub D. and de Montigny M., Preprint CRM-1822, Montral, 1992.

    Google Scholar 

  16. Reid G.J., Euro. J. of Appl. Math., 2, 293 (1991); 2, 319 (1991).

    Google Scholar 

  17. Gazeau J.P. and Winternitz P., Phys. Lett. A167, 246 (1992); J. Math. Phys. 33, 4087 (1992).

    Google Scholar 

  18. Kadomtsev V.V. and Petviashvili V.I., Soy. Phys. Doki. 15, 539 (1970).

    MATH  Google Scholar 

  19. David D., Kamran N., Levi D. and Winternitz P., Phys. Rev. Lett. 55, 2111 (1985); J. Math. Phys. 27, 1225 (1986).

    Google Scholar 

  20. Kac V., Infinite Dimensional Lie Algebras, ( Birkhauser, Boston, 1983 ).

    MATH  Google Scholar 

  21. Kass S., Moody R.V., Patera J., and Slansky R., Affine Lie Algebras, Weight Multiplicities and Branching Rules, (University of California Press, 1990 ).

    MATH  Google Scholar 

  22. Goddard P. and Olive D., Int. J.Mod. Phys. Al, 303 (1986).

    MathSciNet  ADS  Google Scholar 

  23. Champagne B. and Winternitz P., J. Math. Phys, 29, 1 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Martina L. and Winternitz P., Ann. Phys. (N.Y.) 196, 231 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Winternitz P. in Ref. 9.

    Google Scholar 

  26. Jacobson N., Lie Algebras, ( Dover, New York, 1979 ).

    Google Scholar 

  27. Zassenhaus H., Lie Groups, Lie Algebras and Representation Theory, (Presses de l’Universit de Montrai, 1981).

    Google Scholar 

  28. Rand D., Winternitz P. and Zassenhaus H., Lin. Alg. Appl. 109, 197 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  29. Rand D., Comp. Phys. Commun. 41, 105 (1987); 46, 311 (1987).

    Google Scholar 

  30. Rand D., Winternitz P. and Zassenhaus H., Comp. Phys. Commun 46, 297 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Martina L. and Winternitz P., J. Math. Phys. 33, 2718 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Martina L., Soliani G. and Winternitz P., J. Phys. A: Math. Gen. A25, 4425 (1992).

    Article  MathSciNet  ADS  Google Scholar 

  33. Ablowitz M.J., Ramani A. and Segur H., J. Math. Phys. 21, 715 (1980); 21, 1006 (1980).

    Google Scholar 

  34. Rand D. and Winternitz P., Comp. Phys. Commun. 42, 359 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  35. Weiss J., Tabor M. and Carnavale G., J. Math. Phys. 24, 522 (1983).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Levi D. and Winternitz P. (Editors)., Painlevé Transcendents, their Asymptotics and Physical Applications, ( Plenum, New York, 1992 ).

    MATH  Google Scholar 

  37. Dynkin E.B., Amer. Math. Soc. Transl. Ser. 2, 6, 245 (1957).

    Google Scholar 

  38. Cornwell J.F., Group Theory in Physics, Vol. II, (Academic Press, 1984 ).

    MATH  Google Scholar 

  39. Patera J., Sharp R.T., Winternitz P. and Zassenhaus H., J. Math. Phys. 18, 2259 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Beckers J., Hamad J., Perroud M. and Winternitz P., J. Math. Phys. 19, 2126 (1978).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Patera J., Winternitz P. and Zassenhaus H., J. Math. Phys. 16, 1597 (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Patera J., Winternitz P. and Zassenhaus H., J. Math. Phys. 16, 1615 (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Burdet G., Patera J., Perrin M. and Winternitz P., J. Math. Phys. 19, 1758 (1978).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Goursat E., Ann. Sci. Ec. Norm. Sup. (3)., 6, 9 (1889).

    MathSciNet  Google Scholar 

  45. DuVal P., Homographies, Quanternions and Rotations, ( Clarendon Press, Oxford, 1964 ).

    Google Scholar 

  46. Patera J., Sharp R.T., Winternitz P. and Zassenhaus H., J. Math. Phys. 17, 986 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Rubin J., Winternitz P., J. Phys. Math. Gen. A26, 1123 (1993).

    Article  ADS  Google Scholar 

  48. Ndogmo J.C. and Winternitz P., Preprint CRM-1864, Montral,(1993)

    Google Scholar 

  49. Turkowski P., J. Math. Phys. 31, 1344 (1990).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Kruskal M.D., in Ref. 36.

    Google Scholar 

  51. Conte R., in Ref. 36.

    Google Scholar 

  52. Weiss J., in Ref. 36.

    Google Scholar 

  53. Ince E.L., Ordinary Differential Equations, ( Dover, New York, 1956 ).

    Google Scholar 

  54. Painlevé P., Acta Math. 25, 1 (1902).

    Article  MathSciNet  Google Scholar 

  55. Gambier B., Acta Math 33, 1 (1910).

    Article  MathSciNet  Google Scholar 

  56. Cosgrove C. and Scoufis G., Studies Appl. Math 88, 25 (1993).

    MathSciNet  MATH  Google Scholar 

  57. Bureau F., Ann. Mat. Pura Appl. (IV). 91, 163 (1972)

    Article  Google Scholar 

  58. Bureau F., Garcet A. and Goffar J., Ann. Mat. Pura Appl. (IV). 92, 177 (1972), and article in Ref. 36.

    Article  MathSciNet  MATH  Google Scholar 

  59. Bureau F., Acad. Roy. Belg. Bull, Cl. Sc. (5). 73, 335 (1987).

    MathSciNet  MATH  Google Scholar 

  60. Kruskal M.D. and Clarkson P.A., Stud. Appl. Math. 86, 87 (1992).

    MathSciNet  MATH  Google Scholar 

  61. Wang C.S., Phys. Rev. 182, 482 (1969).

    Article  ADS  Google Scholar 

  62. Carman R.L., Shimizu F., Wang C.S. and Bloembergen N., Phys. Rev. A2, 60 (1970).

    ADS  Google Scholar 

  63. Chu F.Y.F. and Scott A.C., Phys. Rev. Al2, 2060 (1975).

    ADS  Google Scholar 

  64. Kaup D.J., Physica 6D, 143 (1983).

    MathSciNet  ADS  Google Scholar 

  65. Kaup D.J. and Menyuk C.R., Phys. Rev. A42, 1712 (1990).

    ADS  Google Scholar 

  66. Levi D., Menyuk C.R. and Winternitz P., Phys. Rev. A44, 6057 (1991).

    MathSciNet  ADS  Google Scholar 

  67. Menyuk C.R., Levi D. and Winternitz P., Phys. Rev. Lett. 69, 3048 (1992).

    Article  ADS  Google Scholar 

  68. Levi D., Menyuk C.R. and Winternitz P., to be published.

    Google Scholar 

  69. Its A.R. and Novokshenov V.Yu., The IsomonodronicDeformation Method in the Theory of PainlevéEquations. ( Springer, Berlin, 1986 ).

    MATH  Google Scholar 

  70. Kitaev A.V., Math. USSR, Sbornik 62, 421 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  71. Menyuk C.R., Phys. Rev. A47, 2235 (1993).

    ADS  Google Scholar 

  72. Byrd P.F. and Friedman M.D., Handbook of Elliptic Integrals for Engineersand Scientists. ( Springer, Berlin, 1971 ).

    Book  MATH  Google Scholar 

  73. Becker D.A. and Richter E.W., Z. Naturforsch. A45, 1219 (1990).

    MathSciNet  Google Scholar 

  74. Meshkov A.G., Soy. Phys. J. 33, 571, (1991).

    Article  MathSciNet  Google Scholar 

  75. Rideau G. and Winternitz P., J. Math. Phys. 31, 1095 (1990).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. Rideau G. and Winternitz P., J. Math. Phys. 34, 558 (1993).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  77. Gagnon L. and Winternitz P., Preprint CRM-1851, Montrai, (1993).

    Google Scholar 

  78. Ibragimov N.H. and Torrisi M., J. Math. Phys. 33, 3931 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  79. Akhatov I.Sh., Gazizov R.K. and Ibragimov N.H., J. Sov. Math. 55, 1401 (1991).

    Article  Google Scholar 

  80. Joshi N., Phys. Lett. Al25, 456 (1987).

    ADS  Google Scholar 

  81. Hlavaty V., Phys. Lett. Al28, 335 (1987).

    MathSciNet  ADS  Google Scholar 

  82. Abellanas L. and Galindo A., Phys. Lett. Al25, 456 (1985).

    Google Scholar 

  83. Clarkson P.A. and Kruskal M.D., J. Math. Phys. 30, 2201 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  84. Levi D. and Winternitz P., J. Phys. A. Math. Gen. 22, 2915 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  85. Bluman G.W. and Cole J.D., J. Math. Mech. 18, 1025 (1969).

    MathSciNet  MATH  Google Scholar 

  86. Olver P. and Rosenau Ph., Phys. Lett. A114 107, (1986).

    MathSciNet  ADS  Google Scholar 

  87. Olver P. and Rosenau Ph., SIAM J. Appl. Math. 47, 263 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  88. Nishitani T. and Tajiri M., Phys. Lett. A89, 179 (1982).

    MathSciNet  Google Scholar 

  89. Lou S.Y., Phys. Lett. A151, 133 (1990).

    ADS  Google Scholar 

  90. Clarkson P.A., J. Phys. A. Math. Gen. 22, 2355 (1989); 22, 3821 (989)

    Google Scholar 

  91. Eur. J. Appl. Math. 1, 279 (1990); Nonlinearity 5, 453, (1992).

    Article  MathSciNet  Google Scholar 

  92. Clarkson P.A. and Winternitz P., Physica 49D, 257 (1991).

    MathSciNet  ADS  Google Scholar 

  93. Clarkson P.A. and Hood S., J. Phys. A.: Math. Gen. 26, 133 (1993).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  94. Nucci M.C. and Clarkson P.A., Phys. Lett. 164, 49 (1992).

    Article  MathSciNet  Google Scholar 

  95. Lou S.Y. and Ni G.J., Commun Math. Phys. 15, 465 (1991).

    Google Scholar 

  96. Pucci E., J. Phys. A.: Math. Gen. 25, 2631 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  97. Pucci E. and Saccomandi G., J. Math. Anal. Appl. 163, 588 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  98. Levi D. and Winternitz P., Phys. Lett. A152, 335 (1991).

    MathSciNet  ADS  Google Scholar 

  99. Levi D. and Rodriguez M.A., J. Phys. A.: Math. Gen. 25, L975 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  100. Quispel G.R.W., Capel H.W. and Sahadevan R., Phys. Lett. A170, 379 (1992).

    MathSciNet  ADS  Google Scholar 

  101. Levi D. and Winternitz P., J. Math. Phys. 34, (1993), to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de recherches mathématiques, Université de Montréal, CP 6128-A, Montréal, H3C 3J7, Québec, Canada

    P. Winternitz

Authors
  1. P. Winternitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain

    L. A. Ibort  & M. A. Rodríguez  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Winternitz, P. (1993). Lie Groups and Solutions of Nonlinear Partial Differential Equations. In: Ibort, L.A., Rodríguez, M.A. (eds) Integrable Systems, Quantum Groups, and Quantum Field Theories. NATO ASI Series, vol 409. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1980-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-1980-1_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4874-3

  • Online ISBN: 978-94-011-1980-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation