Self-duality and Integrable Systems

Abstract

§ 0. IntroductionIn his lectures 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 t P a time variable with respect to P. We study the following evolution equation associated to P:where W=W(x, D^l+^^w^D^e. We denote by <W the set of such operators W. This space *W is a group by the composition of e. We get the KP-hierarchy taking P=D The purpose of this article is to give a foundation for higher dimensional generalization of the KP hierarchy. Let now 6 be the ring of microdifferential operators in several variables. Similarly to the one dimensional case, fixing an operator Pe£, we shall study the following equationCommunicated by M.

Some models in differential geometry

“…Lemma 3.2 (see [9]). Fix an operator P E g. Let a family of D-submodules It satisfy both (3.5) and (3.7) for any t. Then (3.7) reduces to the equation…”

“…In this section we return to the commutative (in the DGA-sense) de Rham complex and describe (following [9]) the Sato equations for deformations of D-modules in some specific example.…”

“…Let us write the equations of motion with respect to the time variables tp and tQ generated by P and Q, (see [9]). The compatibility condition for system (3.9), (3.9') is…”

“…The family of subspaces {E(ra)} m 6 Z defines an increasing filtration of E such that for any ra, n 6 Z Lemma 3.1 (see [9]). Let a D-submodule I satisfy (3.5).…”

“…In w we describe, following [9], the Sato equations for the deformations of D-modules in a certain specific example.…”