Decomposition Of 2+1-dimensional Soliton Equations And Their Quasi-periodic Solutions

In this thesis, our objective is mainly focused on studying the decomposition of 2+1- dimensional soliton equations in continuous and discrete cases and the construction of their quasi-periodic solutions. A decomposition technique is developed to break the 2+1-dimensional continuous and discrete soliton equations into compatible ordinary differential equations or compatible ordinary differential equations and the evolution of discrete flows. With the aid of the solution matrix of Lax equation satisfied by the eigenfunctions, elliptic variables are introduced suitably, which give a direct relation between the soliton equation and the resulting compatible ordinary differential equations. Using the theory of Riemann surfaces and algebraic curves, a unified way is given to construct the Abel-Jacobi coordinates and to straighten out the various flows, both the continuous and the discrete ones. This makes the soliton equations easily?integrated by quadratures. The Riemann- Jacobi inversion is discussed to yield the final expression of explicit solution by means of the Riemann theta function, which is directly related to algebraic geometry. Based on all the results, a general method is developed for solving multidimensional soliton equations, especially, 2+1- and 3+1-dimensional ones. As applications, we obtain quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation, the 2?-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation, the 2?-dimensional modified Korteweg-de Vries equation, a new 3+1-dimensional nonlinear evolution equation, and the 2+1-dimensional Toda lattice equation. Another main attention is paid to the Caos new scheme, which is based on the nonlinearization approach of eigenvalue problems. This new scheme is further shown to be a very powerful tool, through which quasi-periodic solutions of 2+1-dimensional continuous and discrete soliton equations can be obtained. Both methods for the construction of 2-i-i-dimensional integrable models together with techniques for the calculation of exact solutions are discussed. A class of 2+1-dimensional nonlinear evolution equations, the 2+1-dimensional Gardner equation and four new 2+1- dimensional discrete integrable models are decomposed into the 1-i-i-dimensional Jaulent-Miodek equations, new 1+1-dimensional soliton equations, and the 1+1- dimensional nonlinear network equations describing a Volterra system, respectively. Some new finite-dimensional integrable systems and a new integrable symplectic map, a discrete version of classical integrable system, are found through the nonlinearization approach of eigenvalue problems of continuous and discrete types, respectively. These systems itself are of independent interesting. The generating function method is applied to the study of integrability structure of these finite-dimensional systems, by which it is very convenient to prove the involutivity of integrals of motion. Resorting to the elliptic variables and quasi-Abel-Jacobi coordinates, the independence of involutive systems of conserved integrals is proved. These 2+1 -dimensional continuous and discrete integrable models mentioned above are decomposed into the resulting Hamiltonian systems of ordinary differential equations or the Hamiltonian systems of ordinary differential equations plus the discrete flow generated by the symplectic map. In the "window"of the Abel-Jacobi coordinate, a clear evolution pictures of various flows, both continuous and the discrete ones, are presented. All the results enable one to derive quasi-periodic solutions for these 2+1...