Constraint of integrable systems: from higher to lower dimensions

Abstract

We show the constraint which leads the linear system of the n× n matrix KP equation to a (1+1)-dimensional integrable system associated with the symmetric spaces, so that explicit solutions of the matrix KP equation can be derived by solving the resulting system in 1+1 dimensions. In particular the Hermitian solutions obtained in this way correspond to the compact and noncompact real form of the symmetric spaces. The reduction of the matrix KP equation to the matrix KdV and matrix Boussinesq equations reduces the constraint for the latter ones and the resulting system is finite dimensional. The constraint also provides a connection between the matrix KP equation, etc., and the symmetric spaces.