Decomposition Of Soliton Equations And Finite-dimensional Integrable Systems Associated With High-order Spectral Problems

Several high-order continuous spectral problems and a discrete one are studied in the preceding sections of this thesis. Some new soliton hierarchies associated with them are obtained firstly. Then based on these spectral problems and their adjoint ones, suitable constraint relationships (Bargmann constraints or Neumann constraints) between potentials and eigenfunctions are derived through the the nonlinearization approach. Some new finite-dimensional systems (Bargmann systems or Neumann systems) and a new symplectic map (a discrete version of finite-dimensional system) are obtained subsequently. They are further shown to be integrable in the Liouville sense through a technique of generating functions of conserved integrals which generated from Lax matrix. As an application, the calculation of solutions for those hierarchies of soliton equations are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations (for discrete situation, into solving a system of ordinary differential equation plus a simple iterative process of the symplectic map).In the last section, the coupled integrable dispersionless equation is studied. The equation is separated into solvable ordinary differential equations by the aid of corresponding stationary evolution equation and the elliptic coordinates. The hyperelliptic Riemann surface and Abel-Jacobi coordinates are further introduced to straighten out of the associated flows. As an application, the compatible solutions of these flows in Abel-Jacobi coordinates are explicitly obtained.