Solitons and Integrable Systems

Typically wave packets, like an elevation of a water surface, quickly spread and fade away (because of dispersion). Under special circumstances, where a nonlinearity of the medium plays an important role, solitary waves can occur. These are localised excitations keeping their form stable over a relatively long period of time. In the case of water waves in a channel such an observation has been made by the Scottish engineer John Scott Russel (1808-1882) and his subsequent experiments may be regarded as the beginning of soliton physics. In 1877 Boussinesq and in 1895 Korteweg und deVries derived a partial differential equation as a certain limit of the equations of hydrodynamics, as well as an exact solution that correctly displayed the observed properties of a solitary water wave. This equation became famous as the Korteweg-deVries equation, often abbreviated to KdV equation. In suitably scaled variables it takes the form

ut + uxxx + ½ (u2)x = 0

where an index (t or x) indicates a partial derivative with respect to the corresponding coordinate. The notion soliton has been introduced in 1965 by Kruskal and Zabusky. KdV solitons flow through each other and restore their initial form afterwards, only experiencing a displacement in their relative position.

2-soliton KdV solution (animation)

This amazing fact led to enormous activities in mathematics and physics. Soliton structures were detected in many (in particular physical) systems (see, for example, M. Remoissenet, Waves Called Solitons, Springer 1999), new soliton equations emerged, and applications were found (in particular in long range data communication via fiber-optic cables, as modelled by the Nonlinear Schrödinger equation).

Various formalisms have been developed to unravel the underlying mathematical structure. A common feature is the existence of a system of linear equations (Lax pair) such that the respective nonlinear equation arises as its compatibility condition (zero curvature or Zakharov-Shabat condition), and this is usually regarded as the most basic property. Usually closely related to it are many other "integrability features", like infinitely many conservation laws, Bäcklund or Darboux transformations, bi-Hamiltonian structure, extension to a hierarchy (an infinite set of partial differential/difference equations "commuting" with the respective equation).

The word »soliton« is often used more generally in the sense of a (stable, localized) field excitation of finite energy or finite action. Corresponding relatives of the KdV solitons are instantons, kinks, monopoles, vortices, Skyrmions (see R.S. Ward, hep-th/0505135, for a short overview).