• Chou Tian


In the study of nonlinear evolution equations, the concept of symmetry has drawn ever growing attention. It is significant to find the symmetries and their Lie algebra structure for a given equation. In this chapter, the definition of symmetry will be given, and its geometrical meaning will be discussed. Further discussion of concepts and methods related with the finding of symmetry will be given. In order to facilitate the reader’s understanding, the KdV equation will be used as an example through out the text.


Recursion Operator MKdV Equation Galilean Transformation Conservation Quantity Commutator Operation 
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.
    H. H. Chen, Y. C. Lee and G. C. Zhu, Symmetries and Integrability of Cylindrical KdV equation, Plasma Preprint, UMLPF (1984), 85–107.Google Scholar
  2. 2.
    H. H. Chen and Y. C. Lee, On a new hierarchy of symmetries for the Kadomtsefpetviashvilli equations, Physica 9D (1983), 439–445.MathSciNetMATHGoogle Scholar
  3. 3.
    H. H. Chen, Y. C. Lee and G. C. Zhu, Symmetries and Lie algebra for Kadotsef-Petviashvilli„ Plasma Preprint, UMLPF (1984), 85–106.Google Scholar
  4. 4.
    Cheng Yi and Li Yishen, Symmetries and constant of motion for new AKNS hierarchies, J. Phys. A: Math. Gen. 20 (1986), 1951.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheng Yi and Li Yishen, Symmetries and constant of motion for new hierarchy of the K-P equation, Physica D 28 (1986), 189.MathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Daviol, N. Kamran, D. Levi and P. Winternitz, Symmetry reduction for K-P equation using loop algebra, J. Math. Phys. 27 (1986), 1225–1237.MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. S. Fokas and B. Fuchssteiner, On the structure of Sympletic operators and hereditrary symmetries, Lett. Nuovo Cimento 28 (1980), 299–303.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. S. Fokas, A symmetry approach to exactly solvable evolution equations, J. Math. Phys. 21 (1980), 1318–1325.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    A. S. Fokas and R. A. Anderson, On the use of isospectral eigenvalue problem for obtaining hereditary sysmmetries for Hamiltonian system, J. Math. Phys. 22 (1982), 1066–1073.MathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Fuchssteiner, Mastersymmetries, high-order time dependent symmetries and conserved densities of nonlinear evolution equations, Prog. Theo. Phys. 79 (1983), 1508–1522.MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Fuchssteiner and A. S. Fokas, Sympletic structures their Ricklund transformation and hereditary symmetries, Physica 4D (1982), 47–66.MathSciNetGoogle Scholar
  12. 12.
    F. Gonzalez-Gascon, Notes on symmetries of systems of differential equations, J. Math. Phys. 18 (1977), 1763–1767.MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. LLakshmann and K. M. Tamizhmani, Lie Biicklund symmetries of certain nonlinear equations under pertubation around their solution, J. Math. Phys. 26 (1985), 1189–1200.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li Yishen and Zhu G. C., Symmetries of a non-isospectral evolution equation, Chin. Sci. Bull. 19 (1986), 1449–1453.Google Scholar
  15. 15.
    Li Yishen and Zhu G. C., New set of symmetries of the integrable equation, Lie algebra and non-isospectral evolution equations, Scienta, Sinica (1987), 235–244.Google Scholar
  16. 16.
    Li Yishen and Zhu G. C., New set of symmetries of the integrable equations, Lie algebra and non-isospectral evolution equations, J. Phys. A Math. Gen. 19 (1986), 3713–3725.MATHCrossRefGoogle Scholar
  17. 17.
    Li Yishen, Deformation of a class of evolutions, Scientia Sinica 15 (1985), 385–390.Google Scholar
  18. 18.
    Li Yishen and Cheng Yi, Symmetries and constants of motion for new KdV hierarchies, Scientia Sinica 31 (1988), 769.MathSciNetMATHGoogle Scholar
  19. 19.
    M. Leo, R. A. Leo, G. Soliani, L. Solobrino and G. Moucarella, Symmetry properties and bi-Harmiltonian structure of the Toda lattice, Lett. Math. Phys. 8 (1984), 267–272.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Li Y. and He X. B., New symmetries of some nonlinear evolution equations, Preprint Tong Ze University (1986).Google Scholar
  21. 21.
    W. Oevel and A. S. Fokas, Infinitely many commuting symmetries and constant motion for evolution equations, J. Math. Phys. 24 (1984), 918–922.MathSciNetCrossRefGoogle Scholar
  22. 22.
    P. M. Santinian and A. S. Fokas, Symmetries and bi-Hamiltonian structure of 2 + 1 dimensional system, Stud. in Appl. Math. 75 (1986), 179.Google Scholar
  23. 23.
    Tian Chou, Symmetries and a hierarch of the general KdV equation, J. Phys. A Math. Gen. 20 (1987), 359–366.CrossRefGoogle Scholar
  24. 24.
    Tian Chou, New strong symmetry, symmetries and their Lie algebra, Scientia Sinica (Ser. A) 31 (1988), 141–151.MATHGoogle Scholar
  25. 25.
    Tian Chou, Transformation of equations and transformation of symmetries, Acta Mathmaticae Applicatae Sinica 12 (1989), 238–249.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Chou Tian

There are no affiliations available

Personalised recommendations