The Twistor Theory of Equations of KdV Type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to the ^-generalized KdV-hierarchy. It also emerges that each flow of the «-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SLW. It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of the «-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of the n-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space. A summary of the results of the second part of this work appears in the Intro- duction.