Nonlinearization Of A Kind Of Eigenvalue Problem

In this paper, starting with the eigenvalue problem (1.1), the spectral problem related with the vector field {X_n} is turned into a completely integrable Hamilton system on R^2N, with the help of nonlinearization approach. In this process, according to the generating function approach, the involutivity and the functional independence of the conserved integrals are proved, and the Hamiltonian flows are straightened by introducing the Abel-Jacobi coordinates. Then, based on the principles of algebraic curve, the quasi-periodic solutions for the stationary equation (5.7) and the (1+1)-dimentional soliton equation(1.8), (1.9) are obtained by solving the differential equations and inversing the Abel-Jacobi coordinates.