The Spectal Probiem And Its Complete Integrablity Related To A Kind Of Soliton System

In this paper,we discuss the Bargmann system for the complex eigenvalue problem with the energy is dependent on speed Lφ= (a3+aqa-adqx-qxa-ap-pa-r)φ=λφxFirst, some basic concepts are introduced simply. After, based on bi-Hamilton operators K, J and Lenard sequence, by the relation be-tween the potentials (q,p,r) and the eigenvector φ, the assocoated Lax pairs are nonlinearized, then we found the Bargmann system of the eigenvalue problem. According to the Euler-Lagarange function and Legendre transforms, a reasonable Jacobi-Ostrogradsky coordi-nate system has been found. Then the infinite-dimensions Dynamical system can be transformed into the finite-dimendions Hamilton canon-ical system in the symplectic space. Moreover, the representations of the solutions for the evolution equations are generated.