Algebraic analysis of a discrete hierarchy of double bracket equations

Abstract

In this paper we present a discrete version for the hierarchy of double bracket equations \( \tfrac{{\partial L}} {{\partial t_n }} \) = [L, [L, (L n )+]] n = 1, 2, 3, ... where L is a pseudo-differential operator (see [8] to glance at this hierarchy). Our version studies this hierarchy over a space of infinite matrices whose entries are series of power having real coefficients. In this case the Lax operator is obtained dressing the shift matrix Λ which can be considered a discrete analogue of ∂. The structure of the set of all solutions of this new hierarchy is obtained under certain simples algebraic assumptions.