Solution of a discrete inverse scattering problem and of the Cauchy problem of a class of discrete evolution equations

The Cauchy problem of a class of nonlinear evolution equations is solved by finding explicit solutions of a discrete inverse scattering problem that are not restricted to the pure soliton case and implementing appropriate time evolution of the scattering data. This yields operator-valued functions, which are shown to solve a hierarchy of operator evolution equations by applying methods similar to those in Marchenko's work. In addition the relation to canonical Lax constructions is investigated. Using methods introduced by Aden and Carl and Schiebold, one obtains scalar solutions to corresponding scalar equations, sometimes representable by determinants on operator ideals. (C) 1999 American Institute of Physics. [S0022-2488(99)03008-X].