Self-duality and Integrable Systems

Abstract

In his lectures (1984-85) at Kyoto University, Professor M. Sato presented a program for generalizing the soliton theory ([9] ; cf. [10]). The KadomtsevPetviashvili (KP) equation is a typical example of the soliton theory. The KP equation is written in the form of deformation equations of a linear ordinary differential equation. The time evolutions of a solution are interpreted as dynamical motions on an infinite dimensional Grassmann manifold ([7], [9]). The Lie algebra of microdifferential operators of one variable acts on this manifold transitively. He conjectured that any integrable systems can be written in the form of deformation equations of a linear system, and proposed to investigate a deformation of differential equations in higher dimensions. He showed a simple example of a deformation of holonomic systems in higher dimensions ([9]), and its generalization is treated in [4]. In this paper we study a deformation of ^-modules in higher dimensions. First we review the KP equation. We denote by 8 the ring of microdifferential operators of one variable x. We fix a microdifferential operator P, and denote by tP a time variable with respect to P. We study the following evolution equation associated to P: