Some singular motions of non-Abelian Toda lattices

Abstract

Motions of finite Toda lattices are known to be associated with linear wave equations whose general solutions can be expressed in terms of progressing waves, and this association is known to generalize to finite non-Abelian Toda lattices of n x n matrices and systems of n coupled linear wave equations. We present a nontrivial family of non-Abelian Toda lattice motions that can be specialized to ones that are not finite, but not infinitely extendible either, as they contain nonvanishing but singular matrices of rank (n – s). In these cases we give a natural continuation of the lattice dynamics by means of nonsingular matrices of dimension (n – s) x (n – s), and describe how to find s progressing wave solutions of the associated system of n coupled linear wave equations.PACS No.: 5.45-a