The Eigenvalue Problem That A Type Of The Energy Is Dependent On Speed And Its Integerable System

This paper first introduces the related theoretical background and the research status of soliton. Then by defining some basic concepts and theorems to study the third-order eigenvalue problem which the energy is dependent on speed. LΦ=(-(?)3-(?)2q-q(?)2-p(?)+r)=λΦx And its corresponding Bargmann system as well as Hamilton integrable systems under the Bargmann constraint condition.In this paper, by the compatibility condition, the Hamilton operator K、J are established. Based on the Lenart recursive sequence{Gj=1,2,···} and functional gradient, the evolution equations of the eigenvalue problem and its Lax pairs has been found. Thus Bargmann constraint condition has been determined, the equivalent Bargmann system has been given. Afterwards based on the Hamilton mechanics and Euler-Lagrange function a reasonable Jacobi-Ostrongradsky coordinate system in (ω=dφ∧dψ,R6N) has been found, according to this Jacobi-Ostrongradsky coordinate system, the Bargmann system can be equal to the Hamilton canonical system. Finally, the Liouville theorem are applied to prove the integrability of Hamilton canonical system, the involutive representations of the solutions for the nonlineared evolution equation are obtained.