The complex sine-Gordon equation as a symmetry flow of the AKNS hierarchy

Abstract

It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the 'negative' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector nonlinear Schrodinger equations appear as lowest-negative and second-positive flows within the extended hierarchy. This is fully analogous to the well known connection between the sine-Gordon and mKdV equations within the extended mKdV hierarchy. A general formalism for a Toda-like symmetry occupying the 'negative' sector of the sl(N) constrained KP hierarchy and giving rise to the negative Borel sl(N) loop algebra is indicated.