Higher Order Matrix Spectral Problem Of Nonlinear Soliton Equations Of Quasi-periodic Solutions

In recent years, nonlinear science is substantially studied and widely applied in mathematics, physics, chemistry, biology, communication, economics and so on. As a part of nonlinear science, soliton theory is an important branch in mathematics and theoretical physics. Recently, many effective methods have been presented and developed, such as inverse scattering method, B(a|¨)cklund transformation, Darboux transformation, Painlevé analysis, Hirota method and so on, which provide a powerful tool to describe and solve the nonlinear systems.In 1988, Professor Cao developed and proposed a powerful nonlinearization scheme. The main idea is: Under certain constraint between potential functions and eigenfunctions, the corresponding Lax pair is nonlinearized into a finite-dimensional Hamiltonian system. Up to now, three applications of this method has been found as follows:(1) It gives a way to generate new finite-dimensional integrable systems. Given the evolution equation, the integrable system can be obtained by the relevant nonlinear constraints. This judgement has increasingly advanced the research and development of integrable systems.(2) It establishes a bridge between infinite-dimensional systems and finite-dimensional ones. The infinite-dimensional systems can be reduced to finite-dimensional ones by using the technique of the nonlinearization.(3) It provides an effective way to find exact solutions of nonlinear evolution equations.In this thesis, we mainly concern with the high-order matrix spectral problems and the corresponding finite-dimensional integrable systems. Meanwhile, we construct algebro-geometric solution for an extension of AKNS hierarchy and its reduction. The arrangement of this paper is as follows.In Chap 2, we consider a new four-order isospectral problem with three potentials and the corresponding new coupled KdV hierarchy. We consider their generalized bi-Hamiltonian structures via the trace identity. Moreover, a new finite-dimensional Hamiltonian system is produced through the nonlinearization of the Lax pair. Enoughconserved integrals, which are in involution and functionally independent, are created by the Lax operator to guarantee Liouville integrability of the Hamiltonian system.In Chap 3, starting from a matrix spectral problem of 2n-order, we present a hierarchy of nonlinear equations in matrix form. As a special reduction of the hierarchy, a coupled matrix NLS equation and a coupled matrix mKdV equation are found. Under Bargmann constraint between potentials and eigenfunctions, corresponding spectral problem is nonlinearized into finite-dimensional Hamiltonian systems. The Lax representation can be deduced. In the corresponding symplectic manifold, involution and the functional independence of enough conserved integrals are proved. This gives the Liouville integrability of the finite-dimensional Hamiltonian system. Moreover, involutive solutions of the constrained flows are presented.In Chap 4, we consider the quasi-periodic solutions of a new generalized AKNS hierarchy which consists a coupled Schr(o|¨)dinger equation as a special case. Based on finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the equations are separated into solvable ordinary differential equations. Then under the window of Abel-Jacobi coordinate, various flows can be straightened into linear functions of the variables of the flows. By the standard Jacobi inversion treatment, quasi-periodic solutions in terms of the Riemann theta functions of the coupled Schr(o|¨)dinger equation are explicitly constructed. It is worthwhile that the quasi periodic solution of generalized Schr(o|¨)dinger equation is found from that of coupled Schr(o|¨)dinger equations by reduction technique.