SOLITONS AND THE KORTEWEG-DE VRIES
EQUATION: INTEGRABLE SYSTEMS 1834-1995
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
- 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.
- 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  and some brief interpretative discussion of it is
given there .
- See also the several references concerning this `map' (e.g. the Refs
100, 17, 101, 102, 82 and 83) in .
- An earlier version of the `map' printed in double page format is
- 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.
- 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.
- Unfortunately the Ref.  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  will be published
in Acta Applicandae Mathematicae in due course.
- 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 .
- D. J. Gross and A. A. Migdal, Nuclear Physics
B340, 332 (1990).
- 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 .
- Yi Cheng, Yi-Shen Li and R. K. Bullough, J. Phys. A; 21, L443 (1988).
- S. Aoyama and Y. Kodama, Phys. Lett. B278, 1385 (1992).
- 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.