The algebro-geometric solutions for the Fokas–Olver–Rosenau–Qiao (FORQ) hierarchy

Abstract

This paper aims at providing theta function representation of all algebro-geometric solutions and related quantities for the Fokas–Olver–Rosenau–Qiao (FORQ, sometimes also called the modified Camassa–Holm (MCH) in the literature) hierarchy and studying their algebro-geometric initial value problem. Our main tools include the polynomial recursion formalism to derive the FORQ hierarchy, the hyperelliptic curve K n of arithmetic genus n , 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 theta function representations of the Baker–Akhiezer functions, the meromorphic function and the algebro-geometric solutions are obtained for the entire FORQ hierarchy.