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

Zejun Zhou

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 $θ$-function.

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

Abstract

In [14] Dubrovin introduced an -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 -type infinite ODE system, where 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 -type infinite ODE system and Dubrovin's method. As an application, we give a formula for the -point function for the related Riemann -function.
Paper Structure (8 sections, 13 theorems, 187 equations)

This paper contains 8 sections, 13 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. The algebro-geometric $\tau$-functions for the Gelfand--Dickey hierarchy
  3. The ODE system of $W(\lambda)$
  4. Spectral curve and the line bundle of eigenvector
  5. $\theta$-function formulas
  6. Proof of Proposition \ref{['ThmGDtau']}
  7. Applications
  8. Example

Key Result

Proposition 1

The function $Z(t_1,t_2,\dots)$ defined by is the $\tau$-function of the solution to the infinite ODE in InfiODE corresponding to $W(\lambda)$ (and so is a $\tau$-function of the Gelfand--Dickey hierarchy). Here $\theta$ is the $\theta$-function Dubrovin1981Du19Du20AlgCur associated to SpectralCurve, and $t_{a+(n+1)k}:=T^a_k$, $a=1,\dots,n$,

Theorems & Definitions (24)

  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Proposition 2
  • proof
  • ...and 14 more