International Journal of Theoretical Physics

, Volume 36, Issue 6, pp 1329–1339 | Cite as

Multiple singular manifold method and extended direct method: Application to the burgers equation

  • Qu Changzheng


This paper considers the relationship between the multiple singular manifold method (MSMM) and the extended direct method (EDM) for studying partial differential equations. It is shown that the similarity reductions using EDM can be obtained by MSMM. The prototype example for illustrating the approach is the Burgers equation, which is the simplest evolution equation to embody nonlinearity and dissipation. As a conclusion of the MSMM, we obtain a set of Bäcklund transformations of the Burgers equation.


Burger Equation Nonlinear PDEs Singular Manifold Backlund Transformation Conditional Symmetry 


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  1. Arrigo, D. J., Broadbridge, P., and Hill, J. M. (1993).Journal of Mathematical Physics,34, 4692.MATHMathSciNetCrossRefADSGoogle Scholar
  2. Bluman, G. W., and Cole, J. D. (1969).Journal of Mathematics and Mechanics,18, 1025.MATHMathSciNetGoogle Scholar
  3. Chudnovsky, D. V., Chudnovsky, G. V., and Tabor, M. (1983).Physics Letter A,97, 268.MathSciNetCrossRefADSGoogle Scholar
  4. Clarkson, P. A., and Kruskal, M. D. (1989).Journal of Mathematical Physics,30, 2201.MATHMathSciNetCrossRefADSGoogle Scholar
  5. Cole, J. D. (1951).Quarterly Applied Mathematics,9, 225.MATHMathSciNetGoogle Scholar
  6. Estevez, P. G. (1992).Physics Letter A,171, 259.MathSciNetCrossRefADSGoogle Scholar
  7. Estevez, P. G., and Gordoa, P. R. (1995).Studies of Applied Mathematics,95, 73.MATHMathSciNetGoogle Scholar
  8. Estevez, P. G., Gordoa, P. R., Alonso, L. M., and Reus, E. M. (1993).Journal of Physics A: Mathematical and General,26, 1915.MATHMathSciNetCrossRefADSGoogle Scholar
  9. Fokas, A. S. (1979). Ph.D. thesis, California Institute of Technology.Google Scholar
  10. Fokas, A. S., and Liu, Q. M. (1994).Physical Review Letters,73, 3293.MathSciNetCrossRefADSGoogle Scholar
  11. Gordoa, P. R., and Estevez, P. G. (1994).Theoretical Mathematical Physics,34, 4692.Google Scholar
  12. Hood, S. (1995).Journal of Mathematical Physics,36, 1971.MATHMathSciNetCrossRefADSGoogle Scholar
  13. Hopf, E. (1950).Communications in Pure and Applied Mathematics,3, 201.MATHMathSciNetGoogle Scholar
  14. Lie, S. (1881).Archive for Mathematics,6, 328.MATHGoogle Scholar
  15. Nucci, M. C., and Clarkson, P. A. (1992).Physics Letters A,164, 49.MathSciNetCrossRefADSGoogle Scholar
  16. Olver, P. J., and Rosenau, P. (1986).Physics Letters A,114, 107.MATHMathSciNetCrossRefADSGoogle Scholar
  17. Qu, C. Z. (1996).Studies in Applied Mathematics to be published.Google Scholar
  18. Weiss, J. (1983).Journal of Mathematical Physics,24, 1405.MATHMathSciNetCrossRefADSGoogle Scholar
  19. Weiss, J. (1984).Journal of Mathematical Physics,25, 13.MATHMathSciNetCrossRefADSGoogle Scholar
  20. Weiss, J. (1985).Journal of Mathematical Physics,26, 258.MATHMathSciNetCrossRefADSGoogle Scholar
  21. Weiss, J., Tabor, M., and Carnevale, G. (1983).Journal of Mathematical Physics,24, 522.MATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Qu Changzheng
    • 1
  1. 1.Department of MathematicsNorthwest UniversityXi'anChina

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