Completely integrable equations: dynamical groups and their nonlinear realisations

Abstract

It is shown that the dynamical groups of completely integrable equations contain as subgroups the infinite groups of symmetry (of the type Gn infinity ), and the infinite Abelian groups of Backlund transformations. It is demonstrated that the completely integrable equations possess the elementary Backlund transformations of different types. The relation between two methods of describing the completely integrable systems in terms of the field equation and the dynamical group is discussed. It is also shown that the field theories described by the completely integrable equations are the theories of the Nambu-Goldstone type, i.e. these are the theories of spontaneous breakdown of the symmetry with respect to the infinite dynamical groups, and the corresponding fields are the Goldstone ones by which this spontaneous breakdown is accompanied. In particular, any free field can be considered as a Goldstone one.