Spectral Problems Associated With A Class Of Soliton Systems And Their Complete Integrability

This paper focuses on the analysis of a second order spectral problem:L?=((?)2+?(?)u+?v)?=??xand its corresponding Hamilton integrable systemFirstly,Using the compatibility condition of the spectral problem,the double Hamilton operators K and J are obtained,and the evolution equations corresponding to the eigenvalue problem are derived by combining the Lenard recursive sequence.At the same time,the Bargmann constraints are determined by this constraint,and then the Bargmann system under this constraint condition is obtained.The Jacobi-Ostrogradsky coordinate is constructed by the theory of mechanics,and we can get a Hamilton regular system by applying Lax to the nonlinear method.Finally,on the symplectic manifold,the integrable theory is proved to be fully integrable,and the solutions of the evolution equations are obtained.