On a Harry-Dym-Type Hierarchy: Trigonal Curve and Quasi-Periodic Solutions

Resorting to the characteristic polynomial of Lax matrix for a Harry-Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function are introduced. With the help of the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we obtain the quasi-periodic solutions for the entire Harry-Dym-type hierarchy.