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

2026/03/08 by Zejun Zhou · 1 voice
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.AG

paper · pdf

Abstract

In [14] Dubrovin introduced an A₁-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 \mathfrakg-type infinite ODE system, where \mathfrakg 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.

Citations

• Non-crossing permutations for the KP solitons under the Gel'fand-Dickey reductions and the vertex operators
• On an infinite commuting ODE system associated to a simple Lie algebra
• From Toda Hierarchy to KP Hierarchy
• KP solitons and the Riemann theta functions
• Approximating tau-functions by theta-functions
• Gromov--Witten invariants of the Riemann sphere
• Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials
• Simple Lie algebras, Drinfeld--Sokolov hierarchies, and multi-point correlation functions
• Tau Function Approach to Theta Functions
• Geometric interpretation of Zhou's explicit formula for the Witten-Kontsevich tau function
• Explicit Formula for Witten-Kontsevich Tau-Function
• Algebro-Geometric Solutions of the Boussinesq Hierarchy
• Loop groups and equations of KdV type
• A New Approach to the Boussinesq Hierarchy

Discussions

• Zejun Zhou: On algebro-geometric solutions to the Gelfand--Dickey hierarchy https://arxiv.org/abs/2603.07636 https://arxiv.org/pdf/2603.07636 https://arxiv.org/html/2603.07636 [bsky, 0 points, 0 comments]

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