Gauge Theory of Backlund Transformations, I
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.
KeywordsGauge Transformation Half Plane Riccati Equation Jump Condition Nonlinear Schrodinger Equation
Unable to display preview. Download preview PDF.
- 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
- 6.Deift, P. and Trubowitz, E. “Inverse scattering on the line,” Comm. Pure Applied Mathematics, 32, (1979), 121–251.Google Scholar
- 7.Dodd, Eilbeck, Morris, and Gibbons, Boutons and Nonlinear Equation, Academic Press, 1982.Google Scholar
- 8.Flaschka, H. and McLaughlin, D. Some comments on Backlund transformations, and the inverse scattering method,“ in Backlund Transformations, loc. cit.Google Scholar
- 10.Newell, A. Solitons, CAMS, Siam, 1985.Google Scholar
- 11.Nouikov, S., Manakou, S.V., Pitaevskii, L.B., and Zakharov, V.E., Theory of Solitons, Plenum Publishing, New York, 1984.Google Scholar
- 12.Sattinger, D. “Hamiltonian hierarchies on semisimple Lie algebras,” Studies in Appl. Math. 72 (1985), 65–86.Google Scholar
- 14.Zurkowski, V.D. Ph.D thesis, University of Minnesota, June 1987.Google Scholar