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

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 Kn 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.