Infinite Number Of Conservation Laws And Hamiltonian Structure For A Class Of Soliton System

Based on a 2×2 eigenvalue problem, a 3×3 Lenard pair of operators and a new 1+1 dimensional soliton hierarchy are presented. By choosing the special values of parameters in the hierarchy, the hierarchies of the general TD, TD, the general C-KdV and C-KdV can be obtained. In order to investigate the Hamiltonian structure of this soliton hierarchy, a new Lenard gradient sequence {Gj} and its corresponding 2×2 Lenard pair of operators (K, J) are introduced. Further, with the help of Riccati equations, an infinite number of conservation laws for the solton hierarchy are deduced. For the sake of simplicity, taking the general TD hierarchy as an illustrative example, we prove that its 2×2 Lenard pair of operators forms a Hamiltonian pair. Thus the isospectral evolution TD hierarchy is the general Hamiltonian system and possesses the Bi-Hamiltonian structures and Multi-Hamiltonian structures. By using the method of derivation of functional under some constraint condition, a complete one-to-one correspondence between the Hamiltonian functions of the hierarchy and its conservation density functions can be built. These results can also be applied to the isospectral evolution soliton hierarchy of this paper. Finally, there's a gauge transformation between the spectral problem of this paper and the AKNS system. Moreover, the potentials in these spectral problems satisfy the general Miura transformation, the corresponding relationship between the two soliton hierarchies is also given.