Algebro-geometric Constructions to the Dym-type Hierarchy

Resorting to the characteristic polynomial of Lax matrix for the Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function are introduced. Based on 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 get the algebro-geometric constructions for the entire Dym-type hierarch

Trigonal curves and algebro-geometric solutions to soliton hierarchies

Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy