The Energy Of Eigenvalue Problen And Integrability In Bargmann Constraint With The Speed

In this paper. first of all development background and research sta-tus about soliton theory and intergrable system are introduced. Then. some basic concepts are introduced. Using transposition of algebra Lee. the corresponding Bargmann system for the third-order eigenval-ue problem with the energy is dependent on speed Lφ((?)3 +(?)q+p(?)+r)φ=λφxBy means of the compatible condition of main spectral and aux-iliary spectral problem. the bi-Hamilton operators K.J and Lenard sequence are obtained, by the constraint relation between the poten-tials (q,p,r) and the eigenvector φ, the assocoated Lax pairs are non-linearized, then the Bargmann system of the eigenvalue problem is found. Finally, based on the Euler-Lagarange function and Legen-dre transforms, a reasonable Jacobi-Ostrogradsky coordinate system has been established, the Bargmann system is transformed into the Hamilton canonical equations in the coordinate system. Then the infinite-dimensions Dynamical systems can be transformed into the finite-dimendions Hamilton canonical systems in the symplectic space. Moreover, the representations of the solutions for the evolution equa-tions are generated.