Abstract
In a short but remarkable paper Yang and Yang [1] showed that the free energy of a model system consisting of N bosons on a line with repulsive δ-function interactions was given by a set of coupled integral equations. The Yangs’ chosen model is in fact the repulsive version of the quantum Nonlinear Schrödinger (NLS) model. We have shown that with appropriate extensions and different dispersion relations and phase shifts similar formulae apply to ‘all’ of the integrable models quantum or classical. These models include the sine-Gordon (s-G) and sinh-Gordon (sinh-G) models, the two NLS models (attractive and repulsive), the Landau-Lifshitz (L-L’) model which includes all four previous models, and so on. A significant feature is the bose-fermi equivalence of these models: the classical limit of the bose theories yields the classical statistical mechanics of the models in a simple way but in e.g. s-G fermion features remain even in classical limit. The models also include the integrable lattices and the Toda lattice is one of these. We comment on the statistical mechanics of the Toda lattice at the end of the paper. For simplicity the sinh-G model is used throughout this paper to exemplify the method of calculating the free energy, the method of ’generalised Bethe ansatz’.
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References
Yang CN, Yang CP. J.Math.Phys. 10: 115 (1969)
Bullough RK, Caudrey PJ. Solitons. ( Springer, Heidelberg 1980 )
Faddeev LD, Takhtajan LA. Hamiltonian methods in the theory of solitons. ( Springer, Berlin 1987 )
Timonen J, Stirland M, Pilling DJ, Cheng Yi, Bullough RK. Phys.Rev.Lett. 56: 2233 (1986)
Bullough RK, Pilling DJ, Timonen J. J.Phys. A19: L955 (1986)
Timonen J, Chen YZ, Bullough RK. Nucl.Phys. B, 5A: 58 (1988)
Bullough RK, Pilling DJ, Timonen J. In: Lakshmanan M (ed.) Solitons. ( Springer, Heidelberg 1988 )
Timonen J, Pilling DJ, Chen YZ, Bullough RK. In: Bar’yakhtar VG et al. (eds.) Plasma theory and nonlinear and turbulent processes in physics. ( World Scientific, Singapore 1988 )
Bullough RK, Pilling DJ, Timonen J. In: Christiansen PL, Parmentier RD (eds.) Structure, coherence and chaos. ( Manchester UP, Manchester 1988 )
Bullough RK, Timonen J, Chen YZ, Cheng Yi, Stirland M. In: Fordy AP, Degasperis A (eds.) Nonlinear evolution equations: Integrability and spectral methods. ( Manchester UP, Manchester 1989 )
Lieb EH, Liniger W. Phys.Rev. 130: 1605 (1963)
Caudrey PJ. Physica 6D:51 (1982); In: Fordy AP (ed.) Soliton theory a survey of results. ( Manchester UP, Manchester 1989 )
Bullough RK, Pilling DJ, Timonen J. In: Claro (ed.) Nonlinear phenomena in physics ( Springer, Berlin 1985 )
Chen Yu—zhong. Ph.D. Thesis. University of Manchester (submitted March 1989 )
Bullough RK, Olafsson S. In: Solomon A (ed.) Proc. of the 17th Intl. Conf. on Differential Geometric Methods in Theoretical Physics, Chester 1988 ( World Scientific, Singapore 1989 )
Izergin AG, Korepin VE. Lett.Math.Phys. 5: 199 (1981)
Sklyanin EK, Takhtajan LA, Faddeev LD. Teor.Mat.Fiz. 40:194 (1979) [Theor.Math.Phys. 40:688 (1980)]
Thacker HB. Rev.Mod.Phys. 53: 253 (1981)
Cheng Yi. Ph.D. Thesis. University of Manchester (1987)
Flaschka H, McLaughlin DW. Prog.Theor.Phys. 55: 438 (1976)
There is a sign error in Eqns. (7) and (10) of Ref. [5] (k n and k n were accidentally interchanged), but the subsequent equations there are correct
Stone M, Reeve J. Phys. Rev. D18:4746 (1976); Dingle RB, Müller HJW. J. Reine and Angew. Math. 211: 11 (1962)
Bullough RK, Piling DJ, Timonen J. In: Takeno S (ed.) Dynamical problems in soliton systems. ( Springer, Berlin 1985 )
Chen YZ, Bullough RK, Timonen J. To be published
Chen YZ, Bullough RK, Timonen J, Tognetti V, Vaia R. To be published
Fowler M, Zotos X. Phys.Rev. B24:2634 (1982); B25:5806 (1982); Iinada M, Hida K, Ishikawa M. Phys.Lett. 90A:79 (1982); J.Phys. C16: 36 (1983)
Chen NN, Johnson MD, Fowler M. Phys.Rev.Lett. 56:907 (1986); 56:1427(E)(1986); Phys.Rev. B34:1851 (1986); Timonen J, Bullough RK, Pilling DJ. Phys.Rev. B34: 6525 (1986)
Lakshmanan M. Phys.Lett. 61A:53 (1977); Takhtajan LA. Phys.Lett. 64A: 235 (1977)
Bullough RK, Chen YZ, Olafsson S, Timonen J. In: Lochak P, Balabane M, Salem C (eds.) Integrable systems and applications ( Springer, Heidelberg, to appear 1989 )
Nakamura K, Sasada T. J.Phys. C11: L171 (1987)
Toda M. Theory of nonlinear lattices. ( Springer, Berlin 1981 )
Jiang Z, Bullough RK, Manakov SV. Physica 18D:305 (1980); Jiang Zhuhan. Ph.D. Thesis. University of Manchester (1987); Bakurov VG, Manakov SV, Jiang Z, Bullough RK. To be published
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Bullough, R.K., Pilling, D.J., Timonen, J., Cheng, Y., Chen, YZ. (1990). Quantum and Classical Statistical Mechanics of the Integrable Models in 1 + 1 Dimensions. In: Gu, C., Li, Y., Tu, G., Zeng, Y. (eds) Nonlinear Physics. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84148-4_13
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DOI: https://doi.org/10.1007/978-3-642-84148-4_13
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