The algebro-geometric solutions for Hunter-Saxton hierarchy

TL;DR

This paper develops explicit algebro-geometric solutions for the Hunter-Saxton hierarchy using theta functions, hyperelliptic curves, and Baker-Akhiezer functions, advancing the understanding of integrable systems.

Contribution

It provides a comprehensive theta function representation of solutions for the Hunter-Saxton hierarchy, including explicit formulas and key quantities.

Findings

Explicit algebro-geometric solutions for the entire HS hierarchy derived.

Representation of Baker-Akhiezer functions and meromorphic functions obtained.

Connection between solutions and hyperelliptic curves established.

Abstract

This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the Hunter-Saxton (HS) hierarchy through studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the HS hierarchy, 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 Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire HS hierarchy.