Gauge Theory of Backlund Transformations, I

  • D. H. Sattinger
  • V. D. Zurkowski
Part of the NATO ASI Series book series (volume 37)


Backlund transformations are obtained as gauge transformations for Hamiltonian hierarchies over sl(2,C) for potentials in so(2) or su(2). The transformation of the scattering data is calculated, and it is shown how these transformations create or annihilate a pair of eigenvalues in the scattering data, hence create or annihilate a soliton in the potential Q. Repeated Backlund transformations are constructed which introduce higher order poles in the scattering data; and the structure of the higher order singularities is described. It is shown how an arbitrary set of poles in the scattering data may be removed by a sequence of Backlund transformations.


Gauge Transformation Half Plane Riccati Equation Jump Condition Nonlinear Schrodinger Equation 
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  1. 1.
    Beals, R. and Coifman, R., “Scattering and inverse scattering for first order systems, Comm. Pure and Applied Math. 37 (1984), 39–90.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beals, R. and Coifman, R., “Inverse scattering and evolution equations,” Comm. Pure and Applied Math. 38 (1985], 29–42.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Boiti, M., and Tu, G. “Backlund transformations via gauge transformations,” Il Nuovo Cimento, 71B, (19821_ 253–264.MathSciNetGoogle Scholar
  4. 4.
    Chen, H. “Relation between Backlund transformations and inverse scattering problems,” in Backlund Transformations, ed. R.M. Miura, Springer Lecture Notes in Mathematics, #515, Heidelberg, 1976.Google Scholar
  5. 5.
    Darboux, G. C.R. Acad. Sciences, Paris, 94 ( 1882 ], p. 1456.MATHGoogle Scholar
  6. 6.
    Deift, P. and Trubowitz, E. “Inverse scattering on the line,” Comm. Pure Applied Mathematics, 32, (1979), 121–251.Google Scholar
  7. 7.
    Dodd, Eilbeck, Morris, and Gibbons, Boutons and Nonlinear Equation, Academic Press, 1982.Google Scholar
  8. 8.
    Flaschka, H. and McLaughlin, D. Some comments on Backlund transformations, and the inverse scattering method,“ in Backlund Transformations, loc. cit.Google Scholar
  9. 9.
    Flaschka, H. Newell, A., and Ratiu, T. “Kac-Moody Lie algebras and soliton equations,’ Physica 9D (1983), 300–323.MathSciNetMATHGoogle Scholar
  10. 10.
    Newell, A. Solitons, CAMS, Siam, 1985.Google Scholar
  11. 11.
    Nouikov, S., Manakou, S.V., Pitaevskii, L.B., and Zakharov, V.E., Theory of Solitons, Plenum Publishing, New York, 1984.Google Scholar
  12. 12.
    Sattinger, D. “Hamiltonian hierarchies on semisimple Lie algebras,” Studies in Appl. Math. 72 (1985), 65–86.Google Scholar
  13. 13.
    Shabat, A.B. “An inverse scattering problem,” Diff. Equations, 15, (1979), 1299–1307.MathSciNetGoogle Scholar
  14. 14.
    Zurkowski, V.D. Ph.D thesis, University of Minnesota, June 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • D. H. Sattinger
    • 1
  • V. D. Zurkowski
    • 1
  1. 1.University of MinnesotaUSA

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