Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T02:32:56.907Z Has data issue: false hasContentIssue false
  • English
  • Français

On polynomial symmetries of the sine-Gordon equation

Published online by Cambridge University Press:  24 October 2008

G. Z. Tu
G. Z. Tu
Affiliation:
Computing Centre of Chinese Academy of Sciences, Beijing, China
Get access Rights & Permissions [Opens in a new window]

Extract

The sine-Gordon (SG) equation uxt = sin u arises from many branches of physics, and now is one of the most important equations in soliton theory. There have been many works concerning its soliton solutions, Backhand transformations, symmetries and conservation laws and other properties. In this paper we prove that every polynomial symmetry of the SG equation is Hamiltonian, that is, takes the form of D-l δh/δu.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 3 , May 1982 , pp. 485 - 489
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Abellanas, L. and Guil, F.Intrinsic properties for conservation laws and Lie-Backlund invariances of evolution equations. Lett. Math. Phys. 4 (1980), 257–263.CrossRefGoogle Scholar
(2)Dodd, R. K. and Bullough, R. K.Polynomial conserved densities for the sine-Gordon equations. Proc. Roy. Soc. London A 352 (1977), 481–503.Google Scholar
(3)Fordy, A. P. and Gibbon, J.Factorization of operators. I. Miura transformations. J. Math. Phys. 21 (1980), 2508–2510.CrossRefGoogle Scholar
(4)Fuchssteiner, B.Application of hereditary symmetries to nonlinear evolution equations. Nonlinear Analysis, Theory, Method and Appl. 3 (1979), 849–862.CrossRefGoogle Scholar
(5)Guil, F. and Martinez, Alonso L.Generalized variational derivatives in field theory. J. Phys. (A. Math. Gen.) 13 (1980), 689–700.Google Scholar
(6)Kumei, S.On the relationship between conservation laws and invariance groups of non linear field equations in Hamilton's canonical form. J. Math. Phys. 19 (1978), 195–199.CrossRefGoogle Scholar
(7)Lax, P. D.Periodic solutions of the KdV equation. Comm. Pure Appl. Math. 28 (1975), 141–188.CrossRefGoogle Scholar
(8)Lax, P. D. A Hamiltonian approach to the KdV and other equations. In Nonlinear evolution equation, ed. Crandall, M. G. (Academic, New York, 1978).Google Scholar
(9)Mckean, H. P.The sine- and sinh-Gordon equations on a circle. Comm. Pure Appl. Math. 34 (1981), 197–257.CrossRefGoogle Scholar
(10)Newell, A. C.The general structure of integrable evolution equations. Proc. R. Soc. Land. A 365 (1979), 283–311.Google Scholar
(11)Olver, P. J.Evolution equations possessing infinitely many symmetries. J. Math. Phya. 18 (1977), 1212–1215.CrossRefGoogle Scholar
(12)Steudel, H. An infinite set of conservation laws derived by Noether's theorem for several nonlinear evolution equations. In VII Internationale Konferenz uber nichtlineare Schwin-gungen (Band 12, Abhandlung der AdW).Google Scholar
(13)Tu, G. Z.The Lie algebra of invariant group of the KdV, MKdV or Burgers equations. Lett. Math. Phys. 3 (1979), 387–393.Google Scholar
(14)Tu, G. Z. Lax theorem, symmetries and conservation laws. In Symposium on differential geometry and partial differential equations, Beijing, 1980 (Sci. Pr., Beijing, 1981).Google Scholar
(15)Tu, G. Z.Infinitesimal canonical transformations of generalized Hamiltonian equations, J. Math. A. Math. Gen. 15 (1982), 277–285.Google Scholar
(16)Tu, G. Z.On permutability of Backlund transformations. I. Lett. Math. Phys. 6 (1982), 63–71.Google Scholar