arXiv subjects

Zejun Zhou

Publications and source records attributed to Zejun Zhou.

2 recordsLinked to original sources

arXiv2026

On algebro-geometric solutions to the Gelfand--Dickey hierarchy

In [14] Dubrovin introduced an $A_1$-type infinite ODE system and gave a simple way of constructing algebro-geometric solutions to the KdV hierarchy (cf. also [15,4]). In [34] the infinite ODE system is generalized to $\mathfrak{g}$-type infinite ODE system, where $\mathfrak{g}$ is any simple Lie algebra. In this paper, we give a simple constructinon of algebro-geometric solutions to the Gelfand--Dickey hierarchy based on the $A_n$-type infinite ODE system and Dubrovin's method. As an application, we give a formula for the $N$-point function for the related Riemann $\theta$-function.

nlin.SI↗

arXiv2024

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

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 $\tau$-functions. We show that these $\tau$-functions can be identified with the $\tau$-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.

nlin.SI↗