Equivalent forms of multi component Toda hierarchies

In this paper we consider various sets of commuting directions in the Z×ZZ×Z-matrices. For each k≥1k≥1, we decompose the Z×ZZ×Z-matrices in k×kk×k-blocks. The set of basic commuting directions splits then roughly speaking half in a set of directions that are upper triangular w.r.t. this decomposition and half in a collection of directions that possess a lower triangular form. Next we consider deformations of each set in respectively the upper k×kk×k-block triangular Z×ZZ×Z-matrices and the strictly lower k×kk×k-block triangular Z×ZZ×Z-matrices that preserve the commutativity of the generators of each subset and for which the evolution w.r.t. the parameters of the opposite set is compatible. It gives rise to an integrable hierarchy consisting of a set of evolution equations for the perturbations of the basic directions. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent to zero curvature equations for a collection of finite band matrices. As the name zero curvature equations suggests there is a Cauchy problem related to these equations. Therefore a description of the relevant infinite Cauchy problems is given together with a discussion of its solvability and uniqueness. There exists still another form of the nonlinear equations of the hierarchy: the bilinear form. It requires the notion of wave matrices and a description of the related linearizations and then we can show how this bilinear form is equivalent with the Lax form. We conclude with the construction of solutions of the hierarchy.