The Toda Hierarchy and the Kdv Hierarchy

McKean and Trubowitz [2] showed that the theory of the KdV equation ∂ ∂t g(x, t) = ∂ 3 ∂x 3 g(x, t) − 6g(x, t) ∂g ∂x (x, t). is intimately related to the geometry of a related hyperelliptic curve of infinite genus, the Bloch spectrum B g t of the operator L g t : ψ → d 2 dx 2 ψ(x) + g(x, t)ψ(x), where g t = g(x, t). As was known classically, B g t is independent of t, when g(x, t) evolves according to the KdV equation. Our purpose in this paper is to develop a theory of finite difference operators and their Bloch spectra and isospectral flows which mimics the KdV theory. The basic idea of this paper is to use the theory of the periodic Toda chain of length N. Here again, the periodic Toda chain can be understood in terms of a finite genus hyperelliptic curve and isospectral deformations , as van Moerbeke discovered. For instance, see [3]. So one would like to see what the relation of the Toda hierarchy is to the KdV hierarchy, how the conserved quantities are related and so forth. A start on these matters has been obtained by Toda in [4]. In this paper, the idea is that if we choose the initial data for the periodic Toda chain very carefully, then the evolution of this data under the various equations of the Toda hierarchy looks similar to the evolution of f under the KdV hierarchy. Given f, we will find a canonical choice of the initial data of the Toda equations so that the flow of this initial data under the Toda hierarchy looks like the flow of f under the KdV hierarchy, at least to high accuracy. This choice will be given by an asymptotic series in N −1. The main result of this paper is the formulation and partial verification of the Conjecture given below. The method studied here also gives a way of producing analytically approximate solutions to the Toda chain hierarchy, at least conjecturally. My motivation is to use this case as a model for constructing finite genus models for the KP hierarchy, which I have studied in [1]. I also hope to use these methods to develop discrete models of the sine-Gordon equation, the non-linear Schrödinger equation and other infinite dimensional integrable systems that are related to isospectral flows. …