The History of the Early Period of Soliton Theory

Markus Heyerhoff

**Abstract**

The theory of soliton systems today touches an enormous range of theories and methods in various fields of mathematics and allows numerous applications in many areas of natural sciences from physics to biology and engineering. This multidisciplinary character of soliton theory can be traced back to its earliest days. But though soliton theory has different and independant beginnings in fields as far apart as hydrodynamics, solid state physics, field theory, differential geometry and numerics, at the beginning it always focussed more on the soliton than on soliton equations (or even systems of soliton equations). Even in differential geometry the interests of mathematicians were more directed towards the geometric interpretation of (what later became) the solitons: the pseudospherical surfaces. The equations governing these surfaces were of course studied with great care, but more as means to an end. The theory of soliton systems of today therefore started as theory of solitons. That began to change with the connection of the former independant roots of the "early period of soliton theory" and the rapid spreading of the new constituted soliton theory from 1967 on. There the early period ended and classical soliton theory began.

The scientific areas, where solitonic behavior or soliton equations were discovered, can be summarized as follows: The first discoveries can be traced back until 1834. Here the phenomenon of the solitary water wave was investigated, at first experimentally by J. S. Russell [1], later on theoretically too (for example by G. B. Airy [2], Lord Rayleigh [3], M. J. Boussinesq [4] and D. Korteweg and G. de Vries [5]). A complete different branche started in differential geometry in the sixties of the past century. The sine-Gordon equation (sG) was written down (and later solved as well) for the description of surfaces of constant negative curvature. The earliest source known is E. Bour in 1862 [6]. What is known today under the name "Bäcklund transformation" developes just in this context. In 1879 L. Bianchi gave some surface transformation explicit for the sG [7]. S. Lie reformulated it and A. V. Bäcklund gave the transformation its final form by extending it to a one-parameter family [8]. So new solutions can be produced from known solutions; one may even start with the trivial zero-solution. While these branches of the history of soliton theory are more or less well known, a third branch is rather unknown. It started in the physics of crystals, what is known today as solid state physics. The well known physicist J.I. Frenkel and his coworker T.A. Kontorova in Leningrad (USSR) described in 1938 moving dislocations in a crystal by a chain model, that means with continuous time and discretized space variable, which corresponds to the sG [9]. In the early fifties A. Seeger et al. in Stuttgart (Germany) proposed the sG as a model in solid state physics [10]. In particular, they used the Bäcklund transformation to produce exact (solitonic) solutions. But all these branches fell into oblivion. A new idea came from numerics from America. And the discoveries made there had enough penetration power to survive the forgetfulness of man. Numerical studies by E. Fermi, J. Pasta and S. Ulam in 1955 [11] around chain models with nonlinear coupled mass points lead ten years later to the Korteweg-de Vries equation (KdV) [12]. Initiated by the question of integrating the KdV it was only a small step to the development of the inverse scattering method in 1967 [13] and from there to the triumphant advance of soliton theory throughout the world.

References

[1] John Scott Russell: On Waves. Report of the Committee on Waves,... British Ass. for the Adv. of Science, Seventh Report 1837, 417-496.

[2] George Biddell Airy: Tides and waves. Encyclopaedia Metropolitana (1845).

[3] Lord Rayleigh: On Waves. The London, Edinburgh and Dublin
Philosophical Magazine (5) __1__ (1876) 257-279.

[4] M. J. Boussinesq: Théorie des ondes et des remus qui se
propagent le long d´un canal rectangulaire horizontal, ...Journal de
Mathematique Pures et Appliquées (2) __17__ (1872) 55- 108.

[5] Dieterik. J. Korteweg, Gustav de Vries: On the Change of
Form of Long Waves advancing in a Rectangular Canal, and on a New Type of
Stationary Waves. The London, Edinburgh and Dublin Philosophical Magazine __39__
(1895) 422-443.

[6] Edmond Bour: Théorie de la déformation des
surfaces.Journal de l´École Imperiale Polytechnique, __19__ (1862) 1-148.

[7] Luigi Bianchi: Ricerche sulle superficie a curvatura
constante e sulle elicoidi. Tesi di Abilitazione. Annali di Scuola Normale
Superiore Pisa __1,2__ (1879) 285-340.

[8] Albert Victor Bäcklund: Om ytor med konstant negativ
krökning. Lund Universitets Arsskrift __19__, 6 (1883) 1-48.

[9] Jakov Ilitsch Frenkel, Tatjana Abramovna Kontorova: On
the theory of plastic deformation and twinning I, II, III. (In russian) JETP __8__
(1938) 89-95 (I), 1340-1349 (II), 1349-1359 (III).

[10] Alfred Seeger, H. Donth, Albert Kochendörfer: Theorie
der Versetzungen in eindimensionalen Atomreihen III: Versetzungen,
Eigenbewegungen und ihre Wechselwirkungen. Zeitschrift für Physik __134__
(1953) 173-193.

[11] Enrico Fermi, J. Pasta, Stanislaw Martin Ulam: Studies of nonlinear Problems. Los Alamos Scientific Report LA-1940, May 1955.

[12] Norman J. Zabusky, Martin D. Kruskal: Interaction of
"Solitons" in a Collisionless Plasma and the Recurrence of Initial
States. Physical Revew Letters __15,6__ (1965) 240-243.

[13] Clifford S. Gardner, John M. Greene, Martin D. Kruskal,
Robert. M. Miura: Method for Solving the Korteweg-de Vries Equation. Physical
Revew Letters __19__ (1967) 1095-1097.