The Second-order Spectral Problem Associated With Hierarchies Of Evolution Equations And Their Integrability

This paper focuses on a second-order spectral problem where the energy depends on the potential:L?=[?~2-(2p+?)?+?q]?=0.Its corresponding nonlinear evolution equations and Bargmann system are obtained.Under the compatibility condition,the double Hamilton operators K,J and the nonlinear evolution equations corresponding to the spectral problem are calculated.Then the Bargmann constraints are given by the nonlinearization of Lax pairs and Bargmann system corresponding to the spectral problem is constructed.Based on the classical mechanics principle,the generalized momentum is calculated by the Euler-Lagrange equation,then reasonable Jacobi-Ostrogradsky coordinates are introduced in the symplectic space.Under the interaction of Bargmann constraint and Jacobi-Ostrogradsky coordinate system,the corresponding finite-dimensional Hamiltonian canonical system is obtained.Finally,the complete integrability in the sense of Liouville is verified and the solutions of nonlinear evolution equations are obtained.