Integrable hierarchies in the [formula omitted]-matrices related to powers of the shift operator

Abstract

Inside the algebra LTN(R) of N×N-matrices with coefficients from a commutative algebra R over k=R or ℂ, that possess only a finite number of nonzero diagonals above the central diagonal, we consider two deformations of commutative Lie subalgebras generated by the nth power Sn,n⩾1, of the matrix S of the shift operator and a maximal commutative subalgebra h of gln(k), where the evolution equations of the deformed generators are determined by a set of Lax equations, each corresponding to a different decomposition of LTN(R). This yields the h[Sn]-hierarchy and its strict version. We show that both sets of Lax equations are equivalent to a set of zero curvature equations. Next we introduce two Cauchy problems linked with these sets of zero curvature equations and present sufficient conditions under which they can be solved. Moreover, we show that these conditions hold in the formal power series context. Next we introduce two LTN(R)-models, one for each hierarchy, a set of equations in each module and special vectors satisfying these equations from which the Lax equations of each hierarchy can be derived. We conclude by presenting a functional analytic context in which these special vectors can be constructed. Thus one obtains solutions of both hierarchies.