Algebro-geometric solutions for the two-component Camassa-Holm Dym hierarchy

TL;DR

This paper develops explicit algebro-geometric solutions for the two-component Camassa-Holm Dym hierarchy using theta functions, hyperelliptic curves, and Baker-Akhiezer functions, advancing the understanding of integrable systems.

Contribution

It provides explicit theta function representations of solutions for the entire CHD2 hierarchy, utilizing polynomial recursion, hyperelliptic curves, and trace formulas.

Findings

Explicit algebro-geometric solutions for CHD2 hierarchy derived.

Representation formulas involve theta functions and hyperelliptic curves.

Tools include Baker-Akhiezer functions and Dubrovin equations.

Abstract

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Camassa-Holm Dym (CHD2) hierarchy. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire CHD2 hierarchy.