Global solutions of the elliptic 2D periodic Toda lattice

Abstract

This paper demonstrates how to use a dressing construction to produce globally analytic solutions to an elliptic version of the two-dimensional periodic Toda lattice associated with any simple Lie algebra. These equations are seen to occur in the study of harmonic maps and this construction also produces harmonic maps of R2 into a semisimple Lie group or its full flag manifold. In particular, it is shown how to extract Krichever's (1984) theta -function solutions (to the Toda lattice for SLn+1) from the dressing construction. There are harmonic planes in SUn+1 (and its full flag manifold) corresponding to these solutions and their construction is described fairly explicitly.