Tau-functions and dressing transformations for zero-curvature affine integrable equations

Journal of Mathematical Physics

Volumn 38, Issue 2, 1997, Pages 882-901

  Ferreira, Luiz A ^a   Miramontes, J Luis ^b   Guillén, Joaquín Sánchez ^b  

^a SÃO PAULO STATE UNIVERSITY UNESP   (Brazil)

^b UNIVERSITY OF SANTIAGO DE COMPOSTELA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031540337     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531895     Document Type: Article

References (59)

1. R. Hirota, "Direct methods in soliton theory," in Soliton, edited by R. K. Bullough and P. S. Caudrey (Springer, Berlin, 1980), p. 157.
2. R. Hirota, Phys. Rev. Lett. 27, 1192 (1971).
3. T. J. Hollowood, Nucl. Phys. B 384, 523 (1992).
4. H. Aratyn, C. P. Constantinidis, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, Nucl. Phys. B 406, 727 (1993), hep-th/9212086.
5. N. McKay and W. A. McGhee, Int. J. Mod. Phys. A 8, 2791 (1993); Erratum 8, 3830 (1993), hep-th/9208057.
6. M. R. Niedermaier, Commun. Math. Phys. 160, 391 (1994).
7. R. Hirota, J. Phys. Soc. Jpn. 33, 1459 (1972).
8. M. Jimbo and T. Miwa, Publ. RIMS, Kyoto Univ. 19, 943 (1983); E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Publ. RIMS, Kyoto Univ. 18, 1077 (1982).
9. M. Jimbo and T. Miwa, Publ. RIMS, Kyoto Univ. 19, 943 (1983); E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Publ. RIMS, Kyoto Univ. 18, 1077 (1982).
10. V. G. Drinfel'd and V. V. Sokolov, J. Sov. Math. 30, 1975 (1985).
11. G. Wilson, Phil. Trans. R. Soc. London Ser. A 315, 383 (1985); C. R. Acad. Sc. Paris 299 (1), 587 (1984).
12. G. Wilson, Phil. Trans. R. Soc. London Ser. A 315, 383 (1985); C. R. Acad. Sc. Paris 299 (1), 587 (1984).
13. G. Wilson, "The τ-Functions of the gAKNS Equations," in Verdier memorial conference on integrable systems, edited by O. Babelon, P. Cartier, and Y. Kosmann-Schwarzbach (Birkhauser, Basel, 1993), pp. 131-145.
14. V. G. Kac and M. Wakimoto, "Exceptional hierarchies of soliton equations," in Proceedings of Symposia in Pure Mathematics (1989), Vol. 49, p. 191.
15. M. J. Bergvelt and A. P. E. ten Kroode, J. Math. Phys. 29, 1308 (1988).
16. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 13, 166 (1979).
17. M. F. de Groot, T. J. Hollowood, and J. L. Miramontes, Commun. Math. Phys. 145, 57 (1992); N. J. Burroughs, M. F. de Groot, T. J. Hollowood, and J. L. Miramontes, Commun. Math. Phys. 153, 187 (1993); Phys. Lett. B 277, 89 (1992).
18. M. F. de Groot, T. J. Hollowood, and J. L. Miramontes, Commun. Math. Phys. 145, 57 (1992); N. J. Burroughs, M. F. de Groot, T. J. Hollowood, and J. L. Miramontes, Commun. Math. Phys. 153, 187 (1993); Phys. Lett. B 277, 89 (1992).
19. M. F. de Groot, T. J. Hollowood, and J. L. Miramontes, Commun. Math. Phys. 145, 57 (1992); N. J. Burroughs, M. F. de Groot, T. J. Hollowood, and J. L. Miramontes, Commun. Math. Phys. 153, 187 (1993); Phys. Lett. B 277, 89 (1992).
20. T. J. Hollowood and J. L. Miramontes, Commun. Math. Phys. 157, 99 (1993).
21. L. A. Ferreira, J. L. Miramontes, and J. Sánchez Guillén, Nucl. Phys. B 449, 631 (1995) (hep-th/9412127).
22. In Ref. 16, it was shown that the component of A along the central term of g should not be considered as an actual degree of freedom of the hierarchy. This is the reason why these hierarchies can be equivalently formulated both in terms of affine Kac-Moody algebras or of the corresponding loop algebras (see Sec. IV A for more details about this).
23. A. N. Leznov and M. V. Saveliev, Group Theoretical Methods for Integration of Non-Linear Dynamical Systems, Progress in Physics Series, Vol. 15 (Birkhaüser-Verlag, Basel, 1992).
24. L. A. Ferreira, J-L. Gervais, J. Sánchez Guillén, and M. V. Saveliev, "Affine Toda Systems Coupled to Matter Fields," Nucl. Phys. B 470, 236 (1996) (hep-th/9512105).
25. J. W. R. Underwood, "Aspects of Non Abelian Toda Theories," hep-th/9304156.
26. D. Olive and N. Turok, Nucl. Phys. B 257, 277 (1985).
27. D. Olive and N. Turok, Nucl. Phys. B 265, [FS15], 469 (1986).
28. O. Babelon and D. Bernard, Phys. Lett. B 260, 81 (1991); Commun. Math. Phys. 149, 279 (1992) (hep-th/9111036).
29. O. Babelon and D. Bernard, Phys. Lett. B 260, 81 (1991); Commun. Math. Phys. 149, 279 (1992) (hep-th/9111036).
30. i(s), then we have to impose further constraints on the group elements performing the dressing transformation (see Sec. IV B).
31. N among the solutions of the zero-curvature equations.
32. D. Olive, N. Turok, and J. W. R. Underwood, Nucl. Phys. B 401, 663 (1993).
33. D. Olive, N. Turok, and J. W. R. Underwood, Nucl. Phys. B 409, 509 (1993) (hep-th/9305160).
34. D. Olive, M. V. Saveliev, and J. W. R. Underwood, Phys. Lett. B 311, 117 (1993) (hep-th/9212123).
35. O. Babelon and D. Bernard, Int. J. Mod. Phys. A 8, 507 (1993) (hep-th/9206002).
36. N's will be considered as generic changes of variables [see Eqs. (1) and (2)], we will not generally indicate the intrinsic dependence of the tau-functions on the group element h.
37. i(t)〉 [see Eq. (34)].
38. 8," in Symposium on Anomalies, Geometry and Topology, edited by W. A. Bardeen and A. R. White (World Scientific, Singapore, 1985), p. 276.
39. O. Babelon and L. Bonora, Phys. Lett. B 244, 220 (1990); H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, Phys. Lett. B 254, 372 (1991).
40. O. Babelon and L. Bonora, Phys. Lett. B 244, 220 (1990); H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, Phys. Lett. B 254, 372 (1991).
41. T. J. Hollowood, J. L. Miramontes, and Q-Han Park, Nucl. Phys. B 445, 451 (1995) (hep-th/9412062); C. R. Fernńdez-Pousa, M. V. Gallas, T. J. Hollowood, and J. L. Miramontes, "The Symmetric Space and Homogeneous sine-Gordon Theories," to appear in Nucl. Phys. B (hep-th/9606032).
42. T. J. Hollowood, J. L. Miramontes, and Q-Han Park, Nucl. Phys. B 445, 451 (1995) (hep-th/9412062); C. R. Fernńdez-Pousa, M. V. Gallas, T. J. Hollowood, and J. L. Miramontes, "The Symmetric Space and Homogeneous sine-Gordon Theories," to appear in Nucl. Phys. B (hep-th/9606032).
43. A similar construction can be used to connect the solutions of the full generalized mkdV hierarchies with the solutions of the generalized non-Abelian Toda equations of Ref. 20.
44. D. Olive, private communication.
45. L. A. Dickey, "On tau-functions of Zakharov-Shabat and other matrix hierarchies of integrable equations," hep-th/9502063; H. Aratyn, "Integrable Lax Hierarchies, their Symmetry Reductions and Multimatrix Models," lectures at VIII J. A. Swieca Summer School (Rio de Janeiro, Brazil, 1995) (hep-th/9503211).
46. L. A. Dickey, "On tau-functions of Zakharov-Shabat and other matrix hierarchies of integrable equations," hep-th/9502063; H. Aratyn, "Integrable Lax Hierarchies, their Symmetry Reductions and Multimatrix Models," lectures at VIII J. A. Swieca Summer School (Rio de Janeiro, Brazil, 1995) (hep-th/9503211).
47. L. Fehér, J. Harnad, and I. Marshall, Commun. Math. Phys. 154, 181 (1993); L. Fehér and I. Marshall, "Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction, SWAT/95/61," hep-th/9503217; H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, "Constrained KP Models as Integrable Matrix Hierarchies," IFT-P/041/95, UICHEP-TH/95-9, hep-th/9509096.
48. L. Fehér, J. Harnad, and I. Marshall, Commun. Math. Phys. 154, 181 (1993); L. Fehér and I. Marshall, "Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction, SWAT/95/61," hep-th/9503217; H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, "Constrained KP Models as Integrable Matrix Hierarchies," IFT-P/041/95, UICHEP-TH/95-9, hep-th/9509096.
49. L. Fehér, J. Harnad, and I. Marshall, Commun. Math. Phys. 154, 181 (1993); L. Fehér and I. Marshall, "Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction, SWAT/95/61," hep-th/9503217; H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, "Constrained KP Models as Integrable Matrix Hierarchies," IFT-P/041/95, UICHEP-TH/95-9, hep-th/9509096.
50. V. G. Kac, Infinite Dimensional Lie algebras, 3rd ed. (Cambridge U. P., Cambridge, 1990); J. Fuchs, Affine Lie Algebras and Quantum Groups (Cambridge U. P., Cambridge, 1992).
51. V. G. Kac, Infinite Dimensional Lie algebras, 3rd ed. (Cambridge U. P., Cambridge, 1990); J. Fuchs, Affine Lie Algebras and Quantum Groups (Cambridge U. P., Cambridge, 1992).
52. D. H. Peterson and V. G. Kac, Proc. Natl. Acad. Sci. USA 80, 1778 (1983); V. G. Kac, "Constructing groups associated to infinite-dimensional Lie algebras," in Infinite Dimensional Groups with Applications, edited by V. G. Kac, Berkeley MSRI, publication (Springer Verlag, New York, 1985), p. 167.
53. D. H. Peterson and V. G. Kac, Proc. Natl. Acad. Sci. USA 80, 1778 (1983); V. G. Kac, "Constructing groups associated to infinite-dimensional Lie algebras," in Infinite Dimensional Groups with Applications, edited by V. G. Kac, Berkeley MSRI, publication (Springer Verlag, New York, 1985), p. 167.
54. I. B. Frenkel and V. G. Kac, Invent. Math. 62, 28 (1980); J. Lepowsky and R. L. Wilson, Commun. Math. Phys. 62, 43 (1978); V. G. Kac, D. A. Kazhdan, J. Lepowsky, and R. L. Wilson, Adv. Math. 42, 83 (1981); J. Lepowsky, Proc. Natl. Acad. Sci. USA 82, 8295 (1985).
55. I. B. Frenkel and V. G. Kac, Invent. Math. 62, 28 (1980); J. Lepowsky and R. L. Wilson, Commun. Math. Phys. 62, 43 (1978); V. G. Kac, D. A. Kazhdan, J. Lepowsky, and R. L. Wilson, Adv. Math. 42, 83 (1981); J. Lepowsky, Proc. Natl. Acad. Sci. USA 82, 8295 (1985).
56. I. B. Frenkel and V. G. Kac, Invent. Math. 62, 28 (1980); J. Lepowsky and R. L. Wilson, Commun. Math. Phys. 62, 43 (1978); V. G. Kac, D. A. Kazhdan, J. Lepowsky, and R. L. Wilson, Adv. Math. 42, 83 (1981); J. Lepowsky, Proc. Natl. Acad. Sci. USA 82, 8295 (1985).
57. I. B. Frenkel and V. G. Kac, Invent. Math. 62, 28 (1980); J. Lepowsky and R. L. Wilson, Commun. Math. Phys. 62, 43 (1978); V. G. Kac, D. A. Kazhdan, J. Lepowsky, and R. L. Wilson, Adv. Math. 42, 83 (1981); J. Lepowsky, Proc. Natl. Acad. Sci. USA 82, 8295 (1985).
58. A. Fring, P. R. Johnson, M. A. C. Kneipp, and D. I. Olive, Nucl. Phys. B 430, [FS], 597 (1994); M. A. C. Kneipp and D. I. Olive, Commun. Math. Phys. 177, 561 (1996).
59. A. Fring, P. R. Johnson, M. A. C. Kneipp, and D. I. Olive, Nucl. Phys. B 430, [FS], 597 (1994); M. A. C. Kneipp and D. I. Olive, Commun. Math. Phys. 177, 561 (1996).