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


SIGMA 8 (2012), 062, 33 pages      arXiv:1109.1689      https://doi.org/10.3842/SIGMA.2012.062
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Rustem Garifullin a, Ismagil Habibullin a and Marina Yangubaeva b
a) Ufa Institute of Mathematics, Russian Academy of Science, 112 Chernyshevskii Str., Ufa, 450077, Russia
b) Faculty of Physics and Mathematics, Birsk State Social Pedagogical Academy, 10 Internationalnaya Str., Birsk, 452452, Russia

Received April 24, 2012, in final form September 14, 2012; Published online September 18, 2012

Abstract
Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented.

Key words: affine Lie algebra; difference-difference systems; $S$-integrability; Darboux integrability; Toda field theory; integral; symmetry; Lax pair.

pdf (549 kb)   tex (35 kb)

References

  1. Adler V.E., Bobenko A.I., Suris Y.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
  2. Adler V.E., Startsev S.Y., On discrete analogues of the Liouville equation, Theoret. and Math. Phys. 121 (1999), 1484-1495, solv-int/9902016.
  3. Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.
  4. Bobenko A.I., Suris Y.B., Integrable systems on quad-graphs, Int. Math. Res. Not. (2002), 573-611, nlin.SI/0110004.
  5. Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201-209.
  6. Corrigan E., Recent developments in affine Toda quantum field theory, in Particles and Fields (Banff, AB, 1994), CRM Ser. Math. Phys., Springer, New York, 1999, 1-34, hep-th/9412213.
  7. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. II, J. Phys. Soc. Japan 52 (1983), 4125-4131.
  8. Doliwa A., Geometric discretisation of the Toda system, Phys. Lett. A 234 (1997), 187-192, solv-int/9612006.
  9. Drinfel'd V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, J. Math. Sci. 30 (1985), 1975-2036.
  10. Fordy A.P., Gibbons J., Integrable nonlinear Klein-Gordon equations and Toda lattices, Comm. Math. Phys. 77 (1980), 21-30.
  11. Ganzha E.I., Tsarev S.P., Integration of classical series $A_n$, $B_n$, $C_n$, of exponential systems, Krasnoyarsk State Pedagogical University, Krasnoyarsk, 2001.
  12. Garifullin R.N., Gudkova E.V., Habibullin I.T., Method for searching higher symmetries for quad-graph equations, J. Phys. A: Math. Theor. 44 (2011), 325202, 16 pages, arXiv:1104.0493.
  13. Guryeva A.M., Zhiber A.V., On the characteristic equations of a system of quasilinear hyperbolic equations, Vestnik UGATU 6 (2005), 26-34.
  14. Habibullin I.T., Characteristic algebras of fully discrete hyperbolic type equations, SIGMA 1 (2005), 023, 9 pages, nlin.SI/0506027.
  15. Habibullin I.T., Discrete chains of the series $C$, Theoret. and Math. Phys. 146 (2006), 170-–182.
  16. Habibullin I.T., Gudkova E.V., Boundary conditions for multidimensional integrable equations, Funct. Anal. Appl. 38 (2004), 138-148.
  17. Habibullin I.T., Pekan A., Characteristic Lie algebra and the classification of semi-discrete models, Theoret. and Math. Phys. 151 (2007), 781-790, nlin.SI/0610074.
  18. Habibullin I.T., Zheltukhin K., Yangubaeva M., Cartan matrices and integrable lattice Toda field equations, J. Phys. A: Math. Theor. 44 (2011), 465202, 20 pages, arXiv:1105.4446.
  19. Habibullin I.T., Zheltukhina N., Sakieva A., Discretization of hyperbolic type Darboux integrable equations preserving integrability, J. Math. Phys. 52 (2011), 093507, 12 pages, arXiv:1102.1236.
  20. Habibullin I.T., Zheltukhina N., Sakieva A., On Darboux-integrable semi-discrete chains, J. Phys. A: Math. Theor. 43 (2010), 434017, 14 pages, arXiv:0907.3785.
  21. Hirota R., Discrete two-dimensional Toda molecule equation, J. Phys. Soc. Japan 56 (1987), 4285-4288.
  22. Inoue R., Hikami K., The lattice Toda field theory for simple Lie algebras: Hamiltonian structure and $\tau$-function, Nuclear Phys. B 581 (2000), 761-775.
  23. Kimura K., Yamashita T., Nakamura Y., Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices, J. Phys. A: Math. Theor. 44 (2011), 285207, 12 pages.
  24. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215-5266, hep-th/9309137.
  25. Kuniba A., Nakanishi T., Suzuki J., $T$-systems and $Y$-systems in integrable systems, J. Phys. A: Math. Theor. 44 (2011), 103001, 146 pages, arXiv:1010.1344.
  26. Levi D., Yamilov R.I., Generalized symmetry integrability test for discrete equations on the square lattice, J. Phys. A: Math. Theor. 44 (2011), 145207, 22 pages, arXiv:1011.0070.
  27. Leznov A.N., Savel'ev M.V., Group methods for the integration of nonlinear dynamical systems, Nauka, Moscow, 1985.
  28. Leznov A.N., Smirnov V.G., Shabat A.B., Internal symmetry group and integrability conditions for two-dimensional dynamical systems, Theoret. and Math. Phys. 51 (1982), 322-330.
  29. Mikhailov A.V., Integrability of a two-dimensional generalization of the Toda chain, JETP Lett. 30 (1979), 414-418.
  30. Mikhailov A.V., Olshanetsky M.A., Perelomov A.M., Two-dimensional generalized Toda lattice, Comm. Math. Phys. 79 (1981), 473-488.
  31. Nieszporski M., A Laplace ladder diagram of discrete Laplace-type equations, Theoret. and Math. Phys. 133 (2002), 1576-1584.
  32. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, nlin.SI/0110027.
  33. Novikov S.P., Dynnikov I.A., Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds, Russian Math. Surveys 52 (1997), 1057-1116, math-ph/0003009.
  34. Olshanetsky M.A., Perelomov A.M., Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313-400.
  35. Shabat A.B., Higher symmetries of two-dimensional lattices, Phys. Lett. A 200 (1995), 121-133.
  36. Shabat A.B., Yamilov R.I., Exponential systems of type I and the Cartan matrices, Preprint, Bashkirian Branch of Academy of Science of the USSR, Ufa, 1981.
  37. Smirnov S.V., Semidiscrete Toda lattices, arXiv:1203.1764.
  38. Suris Y.B., Generalized Toda chains in discrete time, Leningrad Math. J. 2 (1991), 339-352.
  39. Tsuboi Z., Solutions of discretized affine Toda field equations for $A^{(1)}_n$, $B^{(1)}_n$, $C^{(1)}_n$, $D^{(1)}_n$, $A^{(2)}_n$ and $D^{(2)}_{n+1}$, J. Phys. Soc. Japan 66 (1997), 3391-3398, solv-int/9610011.
  40. Ward R.S., Discrete Toda field equations, Phys. Lett. A 199 (1995), 45-48, solv-int/9502002.
  41. Zabrodin A.V., Hirota's difference equations, Theoret. and Math. Phys. 113 (1997), 1347-1392, solv-int/9704001.

Previous article  Next article   Contents of Volume 8 (2012)