Algebro-Geometric Solutions Of Integrable Hierarchies

Soliton equation, as infinite-dimensional integrable system, has been widely used in both theory and application. Algebro-geometric method is one of the most important tools to study soliton equation. In this paper, algebro-geometric solu-tions and relevant results of four kinds of integrable equations, namely, two inte-grable differential difference hierarchies, relativistic Toda hierarchy and relativistic Lotka-Volterra hierarchy, and two integrable partial differential hierarchies, Fokas-Lenells hierarchy and perturbed Degasperis-Procesi hierarchy, will be considered. In particular, we have derived the N-dark soliton solutions of the whole Fokas-Lenells hierarachy, which can be considered as the completely degenerated case of associated spectral curve. The basic tools involve polynomial recursive formulism, algebraic curve and Riemann theta function.fundamental meromorphic function. Baker-Akhiezer function. Dubrovin-type equation and Lagrange interpolation theo-ry.