R. K. Bullough and P. J. Caudrey, Department of Mathematics, UMIST, PO Box 88, Manchester, M60 1QD, UK

We show in one A3 sheet [1-5] and/or more substantially [6,7] how the soliton, first reported by J. S. Russell in August 1834, has come to occupy, in 1995, a central position in the theory of 2-dimensional quantum gravity. This `A3 sheet' is reproduced here as A4 pages to make it easy to print out, click on LHS and RHS respectively.


  1. R. K. Bullough and J. T. Timonen, Quantum and classical integrable models and statistical mechanics in Proc. 7th Physics Summer School in Statistical Mechanics and Field Theory (Jan 10--28, 1994, Australian National University, Canberra) ed. V. V. Bazhanov (World Scientific Publ. Co. Pte. Ltd., Singapore, 1995). To appear, 1995.

  2. The `map' called `Solitons', which is the `A3 sheet' referred to, is submitted to the publishers (World Scientific) in A3 form but the printed form is to be decided. Copies in A3 size are available from the authors however. This `map' is introduced in § 5 of [1] and some brief interpretative discussion of it is given there [3].

  3. See also the several references concerning this `map' (e.g. the Refs 100, 17, 101, 102, 82 and 83) in [1].

  4. An earlier version of the `map' printed in double page format is in [5].

  5. R. K. Bullough and J. Timonen, in Differential Geometric Methods in Theoretical Physics, Springer Lecture Notes in Physics 375, eds. C. Bartocci, U. Bruzzo and R. Cianci (Springer-Verlag, Heidelberg, 1991) pp. 71-90.

  6. R. K. Bullough and P. J. Caudrey `Solitons and the Korteweg-de Vries Equation - Integrable Systems in 1834-1995' Acta Applicandae Mathematicae 39, 193-228 (1995). The published form of the A3 sheet for the `map' is the single sheet of page 194.

  7. Unfortunately the Ref. [6] as published was not proof-read by the authors and serious printing errors have occurred. The original typescript with the `map' in A3-size and adapted to this text is available from the authors. A correction to [6] will be published in Acta Applicandae Mathematicae in due course.

  8. Two-dimensional quantum gravity is handled explicitly in of [6,7]. The connection derived by Gross and Migdal [6,7,9] and others [6,7] between the functional integral due to Polyakov and the matrix models, and the connection between the one-matrix model and the stationary flows of the whole KdV (Korteweg-de-Vries) hierarchy as equations for the kth `multi-critical point' for the `specific heat' [6,7,9] is given in this §6. The connection between this result and the KP (Kadomtsev-Petviashvili) hierarchy, as well as the KdV hierarchy, via the Weyl algebra as expressed on the `map' is also established in this [10].

  9. D. J. Gross and A. A. Migdal, Nuclear Physics B340, 332 (1990).

  10. For early work on the symmetries of the KP equations and on the algebra in this context see [11,12], and as referenced in [1,6] and especially [13].

  11. Yi Cheng, Yi-Shen Li and R. K. Bullough, J. Phys. A; 21, L443 (1988).

  12. S. Aoyama and Y. Kodama, Phys. Lett. B278, 1385 (1992).

  13. R. K. Bullough and N. M. Bogoliubov, in Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, eds. S. Catto and A. Rocha (World Scientific Publ. Co. Pte. Ltd., Singapore, 1992), 1, 488-504.