An Elementary Introduction to Sato Theory

Abstract

An .elementary introduction to Sato theory is given. Starting with an ordinary differential equation, introducing an infinite number of time variables, and imposing a certain time dependence on the solutions, we obtain the Sato equation which governs the time development of the variable coefficients. It is shown that the generalized Lax equation, the Zakharov-Shabat equation and the IST scheme are generated from the Sato equation. It is revealed that the r-function becomes the key function to express the solutions of the Sato equation. By using the results of the representation theory of groups, it is shown that the r-function is governed by the partial differential equations in the bilinear forms which are closely related to the PlUcker relations.