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


SIGMA 3 (2007), 059, 11 pages      arXiv:0704.3173      http://dx.doi.org/10.3842/SIGMA.2007.059
Contribution to the Vadim Kuznetsov Memorial Issue

A Critical Phenomenon in Solitonic Ising Chains

Igor M. Loutsenko a and Vyacheslav P. Spiridonov b
a) Mathematical Institute, University of Oxford, 24-29 St. Gilles', Oxford, OX1 3LB, UK
b) Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region, 141980 Russia

Received December 04, 2006, in final form April 17, 2007; Published online April 24, 2007

Abstract
We discuss a phase transition of the second order taking place in non-local 1D Ising chains generated by specific infinite soliton solutions of the KdV and BKP equations.

Key words: Ising chain; solitons; phase transition.

pdf (215 kb)   ps (171 kb)   tex (16 kb)

References

  1. Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981.
  2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Math. Appl., Vol. 71, Cambridge Univ. Press, Cambridge, 1999.
  3. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, London, 1982.
  4. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems, World Scientific, Singapore, 1983, 41-119.
  5. Evans M. R., Phase transitions in one-dimensional nonequilibrium systems, cond-mat/0007293.
  6. Gaudin M., Une famille à une paramètre d'ensembles unitaires, Nucl. Phys. 85 (1966), 545-575.
    Gaudin M., Gaz coulombien discret à une dimension, J. Phys. (France) 34 (1973), 511-522.
  7. Hirota R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192-1194.
  8. Kuznetsov V.B., Quadrics on real Riemannian spaces of constant curvature: separation of variables and connection with Gaudin magnet, J. Math. Phys. 33 (1992), 3240-3254.
  9. Loutsenko I.M., Spiridonov V.P., Self-similar potentials and Ising models, Pis'ma v ZhETF (JETP Letters) 66 (1997), 747-753.
    Loutsenko I.M., Spiridonov V.P., Spectral self-similarity, one-dimensional Ising chains and random matrices, Nucl. Phys. B 538 (1999), 731-758.
  10. Loutsenko I.M., Spiridonov V.P., Soliton solutions of integrable hierarchies and Coulomb plasmas, J. Stat. Phys. 99 (2000), 751-767, cond-mat/9909308.
  11. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, 1991.
  12. Spiridonov V.P., Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A 52 (1995), 1909-1935, quant-ph/9601030.
  13. Spiridonov V.P., Zhedanov A.S., Spectral transformation chains and some new biorthogonal rational functions, Comm. Math. Phys. 210 (2000), 49-83.

Previous article   Next article   Contents of Volume 3 (2007)