Finite-dimensional Integrable Systems, Algebraic Geometry Structure Of The Infinite Dimensional Soliton System And Its Explicit Solution

In this thesis we mainly focus on the study of finite-dimensional integrable systems, infinite-dimensional soliton systems and the algebro-geometric construction of their explicit solutions. During the past decades, the nonlinear dynamic system described by soliton equations is substantially studied and widely applied in natural sciences such as biology, chemistry, mathematics, communication and almost all the physical branches like condensed matters, field theory, low temperature physics, hydrodynamics, plasma physics, optics and so on. Meanwhile, finite-dimensional integrable systems are a remarkable class of integrable models in mathematical physics, and a series of excellent examples arose in the development of classical mechanics such as the integrable top of Kovakevski, Euler, and Lagrange, the geodesic flow on an ellipsoid and the harmonic oscillator on a sphere (C. Neumann system). Moreover, it is remarkable to see that these finite-dimensional integrable systems are closely connected with infinite-dimensional integrable systems; that is, most of known finite-dimensional integrable systems could be generated by constraining infinite dimensional integrable systems on the finite dimensional invariant manifolds. To better understand the nonlinear behavior and intrinsic mechanism of practical applications described by soliton equations, it is very important for us to carry out the possible reduction and derive their explicit solutions. For this purpose, we make an endeavor to studying and discussing the following four aspects:Finding New Finite-Dimensional Integrable SystemsIn the theory of integrable systems, it is significant for us to search for as many new finite-dimensional integrable systems as possible. Flaschka pointed out that the reduced systems of integrable systems are integrable as well. Motivated by the Flaschka's idea and Morser's work, in 1989, Professor Cao Cewen originally proposed the nonlinearization of Lax pairs to generate finite-dimensional integrable systems from infinite-dimensional integrable systems (soliton equations). Therefore, combining the Moser's constraint method, the r-matrix theory and the nonlinearization of Lax pairs, we obtain two new Neumann type finite-dimensional integrable systems from the coupled Harry-Dym soliton hierarchy...