The Bargmaim System And Integrable System For A3rd-order Eigenvalue Problem

In this paper, the Bargmann system for the third-order eigenvalue problem with energy dependent on speed, i.e. L(?)=((?)3+2q(?)2+qx(?)+p(?)+(?)p+r)(?)=λ(?)x, is discussed, and the corresponding finite-dimensions Hamilton canonical system are obtained.First, by means of the compatibility conditions (?)xxxt=(?)txxx, the bi-Hamilton operators, i.e. K,J, are gained to find the evolution equations related to the spectral problem and the Lax representation. According to the functional gradient and Lenard recursive sequence, the Bargmann constrained relationship between the potential function (q,p,r) and the eigenvector is obtained and then the associated Lax pairs are nonlineared. Based on the idea of classical mechanics, the system is transformed into an appropriate Lagrange dynamic equation, and then through the discussion of the inverse problem of Lagrange function and finding a group of appropriate Jacobi-Ostrogradsky coordinate, Hamilton canonical equation in the matrix form is thus obtained. Then the complete integrability of its Hamilton system in the Liouville sense is discussed. Finally, the corresponding involutive representations of the solutions to soliton system are generated.