Geometric approach to soliton equations

Abstract

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.