Algebro-Geometric Solutions Of Several Continuous Hierarchies Of Soliton Equations

Solutions of soliton equations not only deeply depict the characteristics of soliton equations and describe the wonderful nonlinear phenomenon, but also help us to understand the essential characteristic of soliton theory. Therefore, the research on the solutions of the soliton equations is a very active research field in soliton theory. Algebra-geometric method is one of the most famous methods appeared on this area, namely using of Riemann surface, Riemann theta function and the inverse problem to solve the algebra-geometric solution(also known as quasi-periodic solution or finite genus solution), which has a profound influence on the development of modern mathematics and theoretical physics.In this paper, under the help of the theory of hyperelliptic curve and algebraic geometry, we construct algebro-geometric solutions of six soliton hierarchies with important physical background. These hierarchies associated with 2 × 2 matrix spectral problems we consider here are Kaup-Newell hierarchy, Heisenberg hierarchy, Coupled Dispersionless hierarchy, Harry Dym hierarchy, WKI hierarchy,Geng hierarchy, respectively.By virtue of the characteristic polynomial of Lax matrix for soliton hiearachy, we define a hyperelliptic curve of arithmetic genus , from which we introduce elliptic variables and the appropriate meromorphic function. Using the kernel of Lenard recursion operators we obtain the asymptotic properties of the meromorphic function at the infinite point and the zero point, respectively. Finally combing these properties of the meromorphic function with their Riemann theta function representations, we get the algebro-geometric solutions of soliton equations. In particular, in the second, third and fourth chapter we straighten the flows of the entire soliton hierarchy under the Abel-Jacobi coordinates.