Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

TL;DR

This paper develops explicit theta function representations for algebro-geometric solutions of the two-component Hunter-Saxton hierarchy, advancing the understanding of its integrable structure and solution space.

Contribution

It introduces a comprehensive method to derive explicit algebro-geometric solutions for the HS2 hierarchy using polynomial recursion, hyperelliptic curves, and Baker-Akhiezer functions.

Findings

Explicit theta function solutions for HS2 hierarchy obtained.

Connection between solutions and hyperelliptic curves established.

Framework applicable to entire HS2 hierarchy.

Abstract

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial value problem. 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 HS2 hierarchy.