Abstract
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.
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References
H. H. Chen, Y. C. Lee and G. C. Zhu, Symmetries and Integrability of Cylindrical KdV equation, Plasma Preprint, UMLPF (1984), 85–107.
H. H. Chen and Y. C. Lee, On a new hierarchy of symmetries for the Kadomtsefpetviashvilli equations, Physica 9D (1983), 439–445.
H. H. Chen, Y. C. Lee and G. C. Zhu, Symmetries and Lie algebra for Kadotsef-Petviashvilli„ Plasma Preprint, UMLPF (1984), 85–106.
Cheng Yi and Li Yishen, Symmetries and constant of motion for new AKNS hierarchies, J. Phys. A: Math. Gen. 20 (1986), 1951.
Cheng Yi and Li Yishen, Symmetries and constant of motion for new hierarchy of the K-P equation, Physica D 28 (1986), 189.
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.
A. S. Fokas and B. Fuchssteiner, On the structure of Sympletic operators and hereditrary symmetries, Lett. Nuovo Cimento 28 (1980), 299–303.
A. S. Fokas, A symmetry approach to exactly solvable evolution equations, J. Math. Phys. 21 (1980), 1318–1325.
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.
B. Fuchssteiner, Mastersymmetries, high-order time dependent symmetries and conserved densities of nonlinear evolution equations, Prog. Theo. Phys. 79 (1983), 1508–1522.
B. Fuchssteiner and A. S. Fokas, Sympletic structures their Ricklund transformation and hereditary symmetries, Physica 4D (1982), 47–66.
F. Gonzalez-Gascon, Notes on symmetries of systems of differential equations, J. Math. Phys. 18 (1977), 1763–1767.
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.
Li Yishen and Zhu G. C., Symmetries of a non-isospectral evolution equation, Chin. Sci. Bull. 19 (1986), 1449–1453.
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.
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.
Li Yishen, Deformation of a class of evolutions, Scientia Sinica 15 (1985), 385–390.
Li Yishen and Cheng Yi, Symmetries and constants of motion for new KdV hierarchies, Scientia Sinica 31 (1988), 769.
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.
Li Y. and He X. B., New symmetries of some nonlinear evolution equations, Preprint Tong Ze University (1986).
W. Oevel and A. S. Fokas, Infinitely many commuting symmetries and constant motion for evolution equations, J. Math. Phys. 24 (1984), 918–922.
P. M. Santinian and A. S. Fokas, Symmetries and bi-Hamiltonian structure of 2 + 1 dimensional system, Stud. in Appl. Math. 75 (1986), 179.
Tian Chou, Symmetries and a hierarch of the general KdV equation, J. Phys. A Math. Gen. 20 (1987), 359–366.
Tian Chou, New strong symmetry, symmetries and their Lie algebra, Scientia Sinica (Ser. A) 31 (1988), 141–151.
Tian Chou, Transformation of equations and transformation of symmetries, Acta Mathmaticae Applicatae Sinica 12 (1989), 238–249.
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© 1995 Springer-Verlag Berlin Heidelberg
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Tian, C. (1995). Symmetry. In: Gu, C. (eds) Soliton Theory and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03102-5_5
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DOI: https://doi.org/10.1007/978-3-662-03102-5_5
Publisher Name: Springer, Berlin, Heidelberg
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