The soliton correlation matrix and the reduction problem for integrable systems

Integrable 1+1 dimensional systems associated to linear first-order matrix equations meromorphic in a complex parameter, as formulated by Zakharov, Mikhailov, and Shabat [1-3] (ZMS) are analyzed by a new method based upon the "soliton correlation matrix" (M-matrix). The multi-Bäcklund transformation, which is equivalent to the introduction of an arbitrary number of poles in the ZMS dressing matrix, is expressed by a pair of matrix Riccati equations for the M-matrix. Through a geometrical interpretation based upon group actions on Grassman manifolds, the solution of this system is explicitly determined in terms of the solutions to the ZMS linear system. Reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing the M-matrix suitably so as to preserve the class of invariant solutions.