Geometric approach to soliton equations

A class of nonlinear equations that can be solved in terms of an $n x n$ scattering problem is investigated. A systematic geometric method of exploiting conservation laws and related equations, the so-called prolongation structure, is worked out. An $n x n$ problem is reduced to n sub $(n - 1) x (n - 1)$ problems and finally to $2 x 2$ problems, which have been comprehensively investigated recently by the author. A general method of deriving the infinite number of polynomial conservation laws for an $n x n$ problem is presented. The $3 x 3$ and $2 x 2$ problems are discussed explicitly.