# Algebro-geometric Constructions to the Dym-type Hierarchy

## TL;DR

This paper develops algebro-geometric methods using trigonal curves and Riemann theta functions to construct solutions for the Dym-type hierarchy, advancing the understanding of integrable systems.

## Contribution

It introduces a novel algebro-geometric framework for the Dym-type hierarchy based on trigonal curves and Baker-Akhiezer functions, providing explicit solution representations.

## Key findings

- Explicit Riemann theta function representations of meromorphic functions.
- Algebro-geometric constructions for the entire Dym-type hierarchy.
- Connection between trigonal curves and integrable systems.

## Abstract

Resorting to the characteristic polynomial of Lax matrix for the Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function are introduced. Based on the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we get the algebro-geometric constructions for the entire Dym-type hierarchy