The algebro-geometric solutions for the modified Camassa-Holm hierarchy

TL;DR

This paper develops explicit theta function representations for algebro-geometric solutions of the modified Camassa-Holm hierarchy, utilizing polynomial recursion, hyperelliptic curves, and Baker-Akhiezer functions.

Contribution

It provides a comprehensive algebro-geometric framework and explicit solutions for the MCH hierarchy using advanced algebraic geometry tools.

Findings

Explicit theta function representations derived

Solutions expressed via hyperelliptic curves and Baker-Akhiezer functions

Crucial quantities for the hierarchy explicitly formulated

Abstract

This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the modified Camassa-Holm (MCH) hierarchy through %and studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the MCH 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 MCH hierarchy.