The Bargmann System For The2^nd-order Eigenvalue Problem And Constraint Flows Of The Soliton Equations

In this paper, the2nd-order eigenvalue problem with the potential Lφ=((?)2-λ2v-λu)φ=αφx is concerned, and the Bargmann system and the evolution equations related to the eigenvalue problem are discussed.First, based on the compatible condition, the reasonable Bi-Hamilton operator K and J are defined, also we obtain the evolution equations related to the eigenvalue problem. Second, by the constraint relation between the potential function and the eigenvector, the Barmann system is obtained, and the Lax pairs of the evolution equations are nonlineared. Based on the Lagrange function and Legendre transforma, a reasonable Jacobi-Ostrogradsky coordinate system has been found. It can be equal to the Hamilton canonical coordinate in real symplecticspace. Finally, the infinite-dimension Dynamical systems can be transformed into the finite-dimension Hamilton canonical systems in the symplectic space, moreover, the Constraint Flows of the Soliton Equations on the finite-dimension subspace are generated.