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


SIGMA 3 (2007), 080, 17 pages      arXiv:0707.1950      http://dx.doi.org/10.3842/SIGMA.2007.080
Contribution to the Vadim Kuznetsov Memorial Issue

Bäcklund Transformation for the BC-Type Toda Lattice

Vadim Kuznetsov a and Evgeny Sklyanin b
a) Deceased
b) Department of Mathematics, University of York, York YO10 5DD, UK

Received July 13, 2007; Published online July 25, 2007

Abstract
We study an integrable case of n-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a Bäcklund transformation and prove its basic properties: canonicity, commutativity and spectrality. The Bäcklund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n+2, which are mutually dual, sharing the same spectral curve.

Key words: Bäcklund transformation; Toda lattice; integrability; boundary conditions; classical Lie algebras.

pdf (295 kb)   ps (179 kb)   tex (19 kb)

References

  1. Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201-209.
  2. Adler M., van Moerbeke P., Kowalewski's asymptotic method, Kac-Moody Lie algebras and regularization, Comm. Math. Phys. 83 (1982), 83-106.
  3. Sklyanin E.K., Boundary conditions for integrable equations, Funktsional. Anal. i Prilozhen. 21 (1987) 86-87 (English transl.: Funct. Anal. Appl. 21 (1987), 164-166).
  4. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  5. Inozemtsev V.I., The finite Toda lattices, Comm. Math. Phys. 121 (1989), 629-638.
  6. Kozlov V.V., Treshchev D.V., Polynomial integrals of Hamiltonian systems with exponential interaction, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 537-556, 671 (English transl.: Math. USSR-Izv. 34 (1990), 555-574).
  7. Olshanetsky M.A., Perelomov M.A., Reyman A.G., Semenov-Tyan-Shansky M.A., Integrable systems. II, in Dynamical Systems. VII. Integrable Systems, Nonholonomic Dynamical Systems, Encyclopaedia of Mathematical Sciences, Vol. 16, Springer-Verlag, Berlin, 1994.
  8. Reyman A.G., Semenov-Tian-Shansky M.A., Integrable systems, Institute of Computer Studies, Moscow, 2003 (in Russian).
  9. Kuznetsov V.B., Sklyanin E.K., On s for many-body systems, J. Phys. A: Math. Gen. 31 (1998), 2241-2251, solv-int/9711010.
  10. Sklyanin E.K., Bäcklund transformations and Baxter's Q-operator, in Integrable Systems: from Classical to Quantum (1999, Montreal), CRM Proc. Lecture Notes, Vol. 26, Amer. Math. Soc., Providence, RI, 2000, 227-250, nlin.SI/0009009.
  11. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer, Berlin, 1987.
  12. Derkachov S.E., Manashov A.N., Factorization of the transfer matrices for the quantum sl(2) spin chains and Baxter equation, J. Phys. A: Math. Gen. 39 (2006), 4147-4159, nlin.SI/0512047.
  13. Veselov A.P., Integrable maps, Russian Math. Surveys 46 (1991), no. 5, 1-51.
  14. Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Birkhäuser, Boston, 2003.
  15. Kuznetsov V.B., Petrera M., Ragnisco O., Separation of variables and Bäcklund transformations for the symmetric Lagrange top, J. Phys. A: Math. Gen. 37 (2004), 8495-8512, nlin.SI/0403028.
  16. Adams M.R., Harnad J., Hurtubise J., Dual moment maps to loop algebras, Lett. Math. Phys. 20 (1990), 294-308.
  17. van Moerbeke P., The spectrum of Jacobi matrices, Invent. Math. 37 (1976), 45-81.
  18. Adler M., van Moerbeke P., Toda-Darboux maps and vertex operators, Int. Math. Res. Not. 10 (1998), 489-511, solv-int/9712016.
  19. Kuznetsov V.B., Salerno M., Sklyanin E.K., Quantum  for DST dimer model, J. Phys. A: Math. Gen. 33 (2000), 171-189, solv-int/9908002.
  20. Kuznetsov V.B., Separation of variables for the Dn type periodic Toda lattice, J. Phys. A: Math. Gen. 30 (1997), 2127-2138, solv-int/9701009.
  21. Pasquier V., Gaudin M., The periodic Toda chain and a matrix generalization of the Bessel function recursion relation, J. Phys. A: Math. Gen. 25 (1992), 5243-5252.
  22. Gerasimov A., Lebedev D., Oblezin S., New integral representations of Whittaker functions for classical Lie groups, arXiv:0705.2886.

Previous article   Next article   Contents of Volume 3 (2007)