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

TL;DR

This paper develops a straightforward method to construct algebro-geometric solutions for the Gelfand--Dickey hierarchy using an $A_n$-type infinite ODE system, extending Dubrovin's approach to a broader class of integrable systems.

Contribution

It introduces a new simple construction of solutions for the Gelfand--Dickey hierarchy based on the $A_n$-type ODE system and Dubrovin's method, expanding the applicability of previous techniques.

Findings

Provides a formula for the N-point function of the associated Riemann theta function.

Extends Dubrovin's method from KdV to Gelfand--Dickey hierarchy.

Simplifies the construction of algebro-geometric solutions for integrable hierarchies.

Abstract

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.