Wronskian Determinants and the Gröbner Representation of a Linear Differential Equation: An Approach to Nonlinear Integrable Systems

Abstract

This chapter presents the theory of integrable systems of nonlinear differential equations. The correspondence between linear differential equations and their solutions gives rise to various types of isomorphisms between Grassmann manifolds and a theory of nonlinear integrable systems are constructed on the bridge where Grassmann manifolds of different origins are linked. The chapter discusses the basic properties of Wronskian determinants in several variables, in the category of Д-modules over a differential field K. In order to interpret the Grassmannian formalism of in explicit terms, the canonical forms of cyclic Д -modules are studied with respect to a well-ordering of the lattice Nr; this gives a generalization of the theory of Gröbner bases for polynomial ideals to a noncommutative case. The chapter also provides a framework for the generalization of a restricted version of the KP hierarchy to higher-dimensional cases.