Non-Abelian integrable systems, quasideterminants and Marchenko

We find explicit (multisoliton) solutions for nonabelian integrable systems such as periodic Toda field equations, Langmuir equations, and Schrödinger equations for functions with values in any associative algebra. The solution for nonabelian Toda field equations for root systems of types A, B, C was expressed by the authors in [EGR] using quasideterminants introduced and studied in [GR1-GR4]. To find multisoliton solutions of periodic Toda equations and other nonabelian systems we use a combination of these ideas with important lemmas which are due to Marchenko [M]. Introduction Let R be an algebra. A map ∂ : R → R is called a derivation of R if ∂(ab) = (∂a)b + a∂b. Suppose that R is an algebra with unit, and ∂ 1 , ∂ 2 two derivations of R commuting with each other. (1) ∂ 1 ((∂ 2 g k)g −1 k) = g k g −1 k−1 − g k+1 g −1 k , k ∈ Z/nZ. In this paper we construct solutions of nonabelian periodic Toda systems and other integrable systems using a combination of the theory of quasideterminants developed in [GR1, GR2, GR3, GR4] and Marchenko approach [M] to integrable systems of differential equations on functions with values in operator algebras. Let Q be a not necessarily commutative algebra with derivations ∂ approach says that when an element Γ ∈ Q satisfies a certain system of linear differential equations connecting the ∂'s, α's and Γ, then its (noncommutative) logarithmic derivative (∂ 1 Γ)Γ −1 satisfies a system of nonlinear equations. Because Marchenko's book is unfortunately unavailable in English, we will reproduce parts of its material for reader's convenience. Note that nonabelian Toda equations for the root system A n−1 were introduced by Polaykov (see [Kr]). Nonabelian Toda lattice for functions of one variable appeared in [PC], [P]. Krichever [Kr] constructed algebraic-geometric solutions for the periodic nonabelian Toda lattices for N × N-matrix-valued functions. Remark. Our construction of multisoliton solutions is similar to the ideas of higher rank solitons proposed by Krichever and Novikov at the end of seventies (see e.g. [KN]).