Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 012, 6 pages      arXiv:0912.2456      https://doi.org/10.3842/SIGMA.2010.012
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

Bäcklund Transformations for the Trigonometric Gaudin Magnet

Orlando Ragnisco and Federico Zullo
Dipartimento di Fisica Universitá Roma Tre and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, I-00146 Roma, Italy

Received December 12, 2009, in final form January 27, 2010; Published online January 29, 2010

Abstract
We construct a Bäcklund transformation for the trigonometric classical Gaudin magnet starting from the Lax representation of the model. The Darboux dressing matrix obtained depends just on one set of variables because of the so-called spectrality property introduced by E. Sklyanin and V. Kuznetsov. In the end we mention some possibly interesting open problems.

Key words: Bäcklund transformations; integrable maps; Gaudin systems.

pdf (168 kb)   ps (128 kb)   tex (10 kb)

References

  1. Bianchi L., Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante, Ann. Sc. Norm. Super. Pisa Cl. Sci. (1) 2 (1879), 285-341.
  2. Bäcklund A.V., Einiges über Curven- und Flächen-Transformationen, Lunds Univ. Årsskr. 10 (1874), 1-12.
  3. Rogers C., Bäcklund transformations in soliton theory, in Soliton Theory: a Survey of Results, Editor A. Fordy, Nonlinear Sci. Theory Appl., Manchester Univ. Press, Manchester, 1990, 97-130.
    Rogers C., Shadwick W.F., Bäcklund transformations and their applications, Mathematics in Science and Engineering, Vol. 161, Academic Press, New York - London, 1982.
  4. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, 1991.
  5. Adler M., On the Bäcklund transformation for the Gel'fand-Dickey equations, Comm. Math. Phys. 80 (1981), 517-527.
  6. Levi D., Nonlinear differential difference equations as Bäcklund transformations, J. Phys. A: Math. Gen. 14 (1981), 1083-1098.
    Levi D., On a new Darboux transformation for the construction of exact solutions of the Schrödinger equation, Inverse Problems 4 (1988), 165-172.
  7. Veselov A.P., What is an integrable mapping?, in What is Integrability?, Editor V.E. Zakharov, Springer Series in Nonlinear Dynamics, Springer, Berlin, 1991, 251-272.
  8. Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, Vol. 219, Birkhäuser, Basel, 2003.
  9. Sklyanin E.K., Separation of variables. New trends, in Quantum Field Theory, Integrable Models and Beyond (Kyoto, 1994), Prog. Theor. Phys. Suppl. (1995), no. 118, 35-60, solv-int/9504001.
  10. Kuznetsov V.B., Sklyanin E.K., On Bäcklund transformations for many-body systems, J. Phys. A: Math. Gen. 31 (1998), 2241-2251, solv-int/9711010.
  11. Kuznetsov V.B., Vanhaecke P., Bäcklund transformations for finite-dimensional integrable systems: a geometric approach, J. Geom. Phys. 44 (2002), 1-40, nlin.SI/0004003.
  12. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the sl(2) Gaudin magnet, J. Phys. A: Math. Gen. 34 (2001), 2477-2490, nlin.SI/0007041.
  13. Hikami K., Separation of variables in the BC-type Gaudin magnet, J. Phys. A: Math. Gen. 28 (1995), 4053-4061.

Previous article   Next article   Contents of Volume 6 (2010)