A New Approach to Noncommutative Soliton Equations

Abstract

A geometric description is introduced to consistently define initial and/or boundary value problem, zero curvature representation and monodromy operator for noncommutative (nc, for short) soliton equations. Based on the description, it is shown that many known commutative/noncommutative soliton equations can be obtained in a unified way. The integrability of a nc-NLS equation and a generalized large N-limit of matrix NLS equation is studied; a generalized conservation law for the nc-NLS equation is presented, and the generalized large N-limit of matrix NLS equation is shown to be integrable in the sense that there exist an infinite number of independent conserved quantities in involution.