Advertisement

Integrability Conditions: Recent Results in the Theory of Integrable Models

  • R. K. Bullough
  • S. Olafsson
  • Yu-Zhong Chen
  • J. Timonen
Part of the NATO ASI Series book series (NSSB, volume 245)

Abstract

This paper reports various results achieved recently in the theory of integrable models. These are summarised in the Fig.1! At the Chester meeting [1] two of the authors were concerned [1] with the local Riemann-Hilbert problem (double-lined box in the centre of Fig.1), its limit as a non-local Riemann-Hilbert problem used to solve classical integrable models in 2+1 dimensions (two space and one time dimensions) [2,3], and the connection of this Riemann-Hilbert problem with Ueno’s [4] Riemann-Hilbert problem associated with the representation of the algebra gl(∞) in terms of Z⊗Z matrices (Z the integers) and the solution of the K-P equations in 2+1. We were also concerned [1] with the construction of the integrable models in 1+1 dimensions from the loop algebras ĝ = g⊗[λ,λ-1] where g is a simple finite dimensional Lie algebra and λ ∈ ℂ. Extensions to super-Lie algebras and super-integrable models in 1+1 were also sketched [1,5,6].

Keywords

Hopf Algebra Quantum Group Symplectic Manifold Monodromy Matrix Quantum Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R K Bullough and S Olafsson, in: “Proc. 17th Intl. Conference on Diff. Geometrical Methods in Theoretical Physics” (Chester Meeting, August 1988) A Solomon ed., World Scientific, Singapore. To appear 1989.Google Scholar
  2. 2.
    Zhuhan Jiang and R K Bullough, J. Phys. A: Math. Gen. 20:L429 (1987).MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3a.
    P J Caudrey, Physica 6D:51 (1982);MathSciNetADSGoogle Scholar
  4. 3a.
    P J Caudrey, in: “Soliton Theory a Survey of Results” A P Fordy ed., Manchester UP, Manchester (1989) and references.Google Scholar
  5. 4.
    Kimio Ueno in: “Vertex Operators in Mathematics and Physics” Math. Sciences Research Institute Publications #3 J Lepowsky, S Mandelstam and I Singer eds, Springer-Verlag, New York (1984) pp.291–302.CrossRefGoogle Scholar
  6. 5.
    R K Bullough, Yu-Zhong Chen, S Olafsson and J Timonen in: “Proc. of the Oléron Colloquium” P Lochak, M Balabane and C Sulem eds, Springer Lecture Notes in Mathematics, Springer-Verlag, Heidelberg (1989). To appear.Google Scholar
  7. 6.
    S Olafsson, J Phys A: Math. Gen. 22:157 (1989).MathSciNetADSMATHCrossRefGoogle Scholar
  8. 7.
    S Olafsson and R K Bullough, and references. To be published (1989).Google Scholar
  9. 8.
    P P Kulish and E K Sklyanin in Proc. Tvärminne Symp., Finland, 1981, J Hietarinta and C Montonen eds, Springer-Verlag, Heidelberg (1982).Google Scholar
  10. 9.
    A G Reyman and M A Semenov-Tian-Shansky, Commun. Maths. Phys. 105:461 (1986).MathSciNetADSMATHCrossRefGoogle Scholar
  11. 10.
    R K Bullough, in: “Solitons” M Lakshmanan ed., Springer Series in Nonlinear Dynamics, Springer-Verlag, Heidelberg (1988) pp.250–281.CrossRefGoogle Scholar
  12. 11.
    J C Eilbeck, J D Gibbon, P J Caudrey and R K Bullough, J Phys. A 6:1337 (1973).ADSCrossRefGoogle Scholar
  13. 12.
    R K Bullough, ‘Solitons’ in: “Interaction of Radiation with Condensed Matter. Vol. I” IAEA-SMR-20/51, Intl. Atomic Energy Agency, Vienna (1977) pp.381–469.Google Scholar
  14. 13.
    R K Bullough, P W Kitchenside, P M Jack and R Saunders, Physica Scripta 20:364 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  15. 14.
    P J Caudrey, J D Gibbon, J C Eilbeck and R K Bullough, Phys. Rev. Letts 30:237 (1973).ADSCrossRefGoogle Scholar
  16. 15.
    P J Caudrey, J D Gibbon, J C Eilbeck and R K Bullough,“Solitons”, Springer Topics in Current Physics 17, R K Bullough and R J Caudrey eds, Springer-Verlag, Heidelberg (1980).Google Scholar
  17. 16.
    R K Bullough and F Ahmad, Phys. Rev. Letts 27:330 (1971).ADSCrossRefGoogle Scholar
  18. 17.
    J Timonen and R K Bullough, Phys. Letts 82A:183 (1981).ADSCrossRefGoogle Scholar
  19. 18.
    R K Bullough, D J Pilling, Yi Cheng, Yu-Zhong Chen and J Timonen, Journal de Physique, Colloque C3. Suppl. au n°3, 50:C3–41 (1989).Google Scholar
  20. 19.
    R K Bullough, J Timonen, Yu-Zhong Chen, Yi Cheng and M Stirland in: “Proc. Intl. Workshop on Nonlinear Evolution Equations: integrability and spectral methods” (Como, Italy, July 1988) A P Fordy and A Degasperis eds, Manchester UP, Manchester (1989).Google Scholar
  21. 20.
    G T Zimanyi, S A Kivelson and A Luther, Phys. Rev. Letts 60:2089 (1988).ADSCrossRefGoogle Scholar
  22. 21.
    C Itzykson and J B Zuber, “Quantum Field Theory”, McGraw-Hill, New York (1980).Google Scholar
  23. 22.
    R K Bullough, in: “Nonlinear Phenomena in Physics” F Claro ed., Springer-Verlag, Heidelberg (1989) pp.70–102.Google Scholar
  24. 23.
    R P Feynman and A R Hibbs, “Quantum Mechanics and Path Integrals”, McGraw-Hill, New York (1965).MATHGoogle Scholar
  25. 24.
    H W Braden, E Corrigan, P E Dorey and R Sasaki, “Aspects of Perturbed Conformal Field Theory, Affine Toda Field Theory and Exact S-Matrices”. Paper this meeting.Google Scholar
  26. 25.
    R K Dodd and R K Bullough, Proc. Roy. Soc. London A 352:481 (1977).MathSciNetADSCrossRefGoogle Scholar
  27. 26.
    P Christe, “S-Matrices of the Tricritical Ising Model and Toda System”. Paper this meeting.Google Scholar
  28. 27.
    L D Faddeev and L A Takhtajan, “Hamiltonian Methods in the Theory of Solitons”, Springer-Verlag, Heidelberg (1987).MATHCrossRefGoogle Scholar
  29. 28.
    J Liouville, Journal de Math. 20:137 (1855).Google Scholar
  30. 29.
    V I Arnold, “Mathematical Methods of Classical Mechanics”, Springer-Verlag, Heidelberg (1978).MATHGoogle Scholar
  31. 30.
    E Witten, in: “IXth Intl. Congress on Math. Phys.” B Simon, A Truman and I M Davies eds, Adam Hilger, Bristol (1989).Google Scholar
  32. 31.
    Yong-Shi Wu, “Topological Chern-Simons Gauge Theories and New Hierarchies of Knot/Link Polynomials”. Paper this meeting.Google Scholar
  33. 32.
    R J Baxter, “Exactly Solved Models in Statistical Mechanics”, Academic Press, New York (1982).MATHGoogle Scholar
  34. 33.
    M Wadati and Y Akutsu, in: “Solitons” Springer Series in Nonlinear Dynamics, M Lakshmanan ed. Springer-Verlag, Heidelberg (1988) pp.282–306.CrossRefGoogle Scholar
  35. 34.
    Vaugh Jones, “Baxterization”. Paper this meeting.Google Scholar
  36. 35.
    S Saito, Phys. Rev. D 36:1819 (1987).MathSciNetADSCrossRefGoogle Scholar
  37. 36.
    M Mulase, J. Diff. Geom. 19:403 (1984).MathSciNetMATHGoogle Scholar
  38. 37.
    R K Bullough, Yu-Zhong Chen and J Timonen. To be published.Google Scholar
  39. 38.
    G Venturi, “Quantum Gravity and the Berry Phase”. Paper this meeting.Google Scholar
  40. 39.
    J Timonen, M Stirland, D J Pilling and R K Bullough, Phys. Rev. Letts 56:2233 (1986).MathSciNetADSCrossRefGoogle Scholar
  41. 40.
    R Sasaki and R K Bullough in: “Nonlinear Evolution Equations and Dynamical Systems” Lecture Notes in Physics 120, M Boiti, F Pempinelli and G Soliani eds, Springer-Verlag, Heidelberg (1980) pp.315–337.Google Scholar
  42. 41.
    R Sasaki and R K Bullough, Proc. Roy. Soc. London A, 376:401 (1981).MathSciNetADSMATHCrossRefGoogle Scholar
  43. 42.
    E Sklyanin, LOMI Preprint E-3–1979 (1979).Google Scholar
  44. 43.
    Z Jiang, “Integrable Systems and Integrability”, PhD Thesis, University of Mancheseter (Feb. 1987).Google Scholar
  45. 44.
    M Jimbo, Lett. Math. Phys. 10:63 (1985).MathSciNetADSMATHCrossRefGoogle Scholar
  46. 45a.
    V G Drinfeld, Soviet Math. Dokl. 27 (No.l):68 (1983);MathSciNetGoogle Scholar
  47. 45a.
    V G Drinfeld, “Quantum Groups” in: Proc. Intl. Cong, of Mathematicians, Berkeley (1987) pp.789–820.Google Scholar
  48. 46.
    S Majid, “Quasitriangular Hopf Algebras and the Yang-Baxter Equations”, Int. J. Mod. Phys. A (1989). To appear.Google Scholar
  49. 47.
    Yu I Manin, Ann. Inst. Fourier, Grenoble 37, 4:191 (1987).MathSciNetMATHCrossRefGoogle Scholar
  50. 48.
    I Paczek, “Statistical Mechanics of Some Integrable Quantum Spin Systems in One Dimension”, PhD Thesis, University of Manchester. To be submitted (1990).Google Scholar
  51. 49.
    M Jimbo and T Miwa, “Integrable Systems and Infinite Lie Algebras” in: “Integrable Systems in Statistical Mechanics”, G M D’Ariano, A Montorsi and M G Rasetti eds, World Scientific, Singapore (1985).Google Scholar
  52. 50.
    H B Thacker, Rev. Mod. Phys. 53:253 (1981).MathSciNetADSCrossRefGoogle Scholar
  53. 51.
    Yi Cheng, “Theory of Integrable Lattices”, PhD Thesis, University of Manchester (Jan. 1987).Google Scholar
  54. 52.
    A G Izergin and V E Korepin, Lett. Math. Phys. 5:199 (1981).MathSciNetADSCrossRefGoogle Scholar
  55. 53.
    Yu-Zhong Chen, “Statistical Mechanics of the Classical and Quantum 1+1 Dimensional Integrable Models”, PhD Thesis, University of Manchester (July 1989).Google Scholar
  56. 54.
    R K Bullough, D J Pilling and J Timonen, J. Phys. A: Math. Gen. 19:L955 (1986).MathSciNetADSMATHCrossRefGoogle Scholar
  57. 55.
    C N Yang and C P Yang, J. Math. Phys. 10:1115 (1969).ADSMATHCrossRefGoogle Scholar
  58. 56.
    J Timonen, Yu-Zhong Chen and R K Bullough, Nucl. Phys. B (Proc. Suppl.) 5A:58 (1988).ADSCrossRefGoogle Scholar
  59. 57a.
    Nu Nu Chen, M D Johnson and M Fowler, Phys. Rev. Letts 56: 507 (1986);Google Scholar
  60. 57b.
    Nu Nu Chen, M D Johnson and M Fowler, Phys. Rev. Letts 56: 1427 (E) (1986).ADSCrossRefGoogle Scholar
  61. 58.
    J Timonen, R K Bullough and D J Pilling, Phys. Rev. B 34:6525 (1986).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • R. K. Bullough
    • 1
  • S. Olafsson
    • 1
  • Yu-Zhong Chen
    • 1
  • J. Timonen
    • 2
  1. 1.Department of MathematicsUMISTManchesterUK
  2. 2.Physics DepartmentUniversity of JyväskyläJyväskyläFinland

Personalised recommendations