The Toda hierarchy and the KdV hierarchy

Abstract

The Toda hierarchy of size $N$ is well known to be analogous to the KdV hierarchy at $N$ goes to infinity. This paper shows that given $f$ a periodic function, there is a canonical way of defining the initial data for the Toda lattice equations so that the evolution of this data under the Toda lattice hierarchy looks asymptotically like the evolution of $f$ under the KdV hierarchy. Further, the conserved quantities of $f$ and those of the Toda hierarchy match.