Algebro-Geometric Solutions of the TD Hierarchy

Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a 2 x 2 matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function I center dot, also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve . The known zeros and poles of I center dot allow to find theta function representations for I center dot by referring to Riemann's vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of I center dot and its theta function representation.