ar X iv : q - a lg / 9 70 70 17 v 1 1 4 Ju l 1 99 7 NONABELIAN INTEGRABLE SYSTEMS , QUASIDETERMINANTS , AND MARCHENKO LEMMA

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.