A Hierarchy Of Evolution Equations And Its Integrable System

In this paper,we discuss the second-order matrix eigenvalue problem:φ_x=M_φ, the evolution equations and its Lax pairs associated with second-order matrix eigenvalue problem, then based on the Bargmann constraint between the potential and the eigenfunctions, a new finitedimensional Hamiltonian system is obtained by nonlinearzation of the eigenvalue problem. It is proved to be that the new finite-dimensional Hamiltonian system is completely integrable in the Liouville sense. Finally based on the involutive solution of completely integrable Hamiltonian system in the Liouville sense involution representation of the solution for the evolution equations are generated.