On an infinite commuting ODE system associated to a simple Lie algebra

Abstract

Inspired by a recent work of Dubrovin [7], for each simple Lie algebra $\mathfrak{g}$, we introduce an infinite family of pairwise commuting ODEs and define their $\ddot{I}„$-functions. We show that these $\ddot{I}„$-functions can be identified with the $\ddot{I}„$-functions for the Drinfeld--Sokolov hierarchy of $\mathfrak{g}$-type. Explicit examples for $\mathfrak{g}=A_1$ and $A_2$ are provided, which are connected to the KdV hierarchy and the Boussinesq hierarchy respectively.