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Conservation laws and symmetries of generalized sine-Gordon equations

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Abstract

We study some systems of non-linear PDE's (Eqs. 1.1 below) which can be regarded either as generalizations of the sine-Gordon equation or as two-dimensional versions of the Toda lattice equations. We show that these systems have an infinite number of non-trivial conservation laws and an infinite number of symmetries. The second result is deduced from the first by a variant of the Hamiltonian formalism for evolution equations. We also consider some specializations of the systems.

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References

  1. Ablowitz, M. J., Kaup, D. J., Newell, A. C., Segur, H.: Stud. Appl. Math.53, 249–315 (1974)

    Google Scholar 

  2. Bogoyavlensky, O. I.: Commun. Math. Phys.51, 201–209 (1976)

    Google Scholar 

  3. Dodd, R. K., Bullough, R. K.: Proc. R. Soc. Lond. A352, 481–503 (1977)

    Google Scholar 

  4. Fordy, A. P., Gibbons, J.: Commun. Math. Phys.77, 21–30 (1980)

    Google Scholar 

  5. Gel'fand, I. M., Manin, Yu. I., Shubin, M. A.: Funct. Anal. Appl.10: 4, 30–34 (Russian); 274–278 (English) (1976)

  6. Kupershmidt, B. A., Wilson, G.: Invent. Math.62, 403–436 (1981)

    Google Scholar 

  7. Lax, P. D.: Commun. Pure Appl. Math.21, 467–490 (1968)

    Google Scholar 

  8. Lax, P. D.: Adv. Math.16, 368–379 (1975)

    Google Scholar 

  9. Mikhailov, A. V.: Pis'ma Zh. Eksp. Teor. Fiz.30, 443–448 (1979); JETP Lett.30, 414–418 (1979)

    Google Scholar 

  10. Mikhailov, A. V., Olshanetsky, M. A., Perelomov, A. M.: Commun. Math. Phys.79, 473–488 (1981)

    Google Scholar 

  11. Miura, R. M., Gardner, C. S., Kruskal, M. D.: J. Math. Phys.9, 1204–1209 (1968)

    Google Scholar 

  12. Wilson, G.: Math. Proc. Cambridge Philos. Soc.86, 131–143 (1979)

    Google Scholar 

  13. Zhiber, A. V., Shabat, A. B.: Dokl. Akad. Nauk SSSR247, 1103–1107 (1979); Sov. Phys. Dokl.24, 607–609 (1979)

    Google Scholar 

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Authors and Affiliations

  1. University of Michigan, 48109, Ann Arbor, MI, USA

    B. A. Kupershmidt

  2. Mathematical Institute, 24-29 St Giles, OX1 3LB, Oxford, England

    George Wilson

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  1. B. A. Kupershmidt
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  2. George Wilson
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Communicated by J. Moser

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Kupershmidt, B.A., Wilson, G. Conservation laws and symmetries of generalized sine-Gordon equations. Commun.Math. Phys. 81, 189–202 (1981). https://doi.org/10.1007/BF01208894

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  • DOI: https://doi.org/10.1007/BF01208894

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