Lie groups, spin equations, and the geometrical interpretation of solitons

Abstract

The integrable evolution equations imbeddable in SU (2) are shown to have two gauge equivalent forms; the AKNS form, and a spin form for which the field is a three-dimensional vector of unit length. These equations are the compatibility conditions for the existence of a bilocal Lie group in two distinct frames of reference. These frames are associated with moving bases on surfaces formed by the motion of the strings introduced by Lamb. Both forms of the evolution equation are derivable from a locality assumption for the generators of the bilocal Lie group. The assumption is sufficient to distinguish between integrable and nonintegrable systems imbedded in SU (2).