The Involutive Solutions Of The Derivative Nonlinear Schr(?)dinger Hierarchy

With the continuous improvement and development of science and technology,soliton and integrable system are seen in many fields.Soliton-related problems can be found in biology,fluids,physics,optical fiber communications,medicine and geology.In order to expand the application range of the soliton,it is necessary to strengthen the in-depth study of the theory of the soliton and the integrable system and enrich the content.In this paper,the Bargmann system for the 2th-order spectral problem:(?) associated with derivative nonlinear Schrodinger hierarchy is discussed.Based on the equispectral compatibility condition,the Lenard recursive operators K,J are given,and the corresponding nonlinear evolution equations of the spectral problem are obtained.Under the relationship between the potential function(u,v)and the characteristic function ?,the nonlinearization of Lax pair for the corresponding nonlinear evolution equations are carried out,and then the Bargmann system for spectral problems are constructed.According to the viewpoint of Hamiltonian mechanics,a reasonable Jacobi-Ostrongradsky coordinate system is obtained.By means of Liouville theorem,the complete integrability of Hamilton canonical system in Liouville sense is proved,and the involute representation of solutions of derivative nonlinear Schrodinger hierarchy is obtained.