A Soliton Hierarchy Associated With A 2×2 Matrix Spectral Problem And Their Infinite-Dimensional Bi-Hamiltonian Structure

Based on a 2 x 2 spectral problem, a 3 x 3 Lenard pair of operators and a new (1+1) dimen-sional soliton hierarchy are presented. In order to investigate the Hamiltonian structures of this soliton hierarchy, a new Lenard gradient sequence Gj and its corresponding 2×2 Lenard pair of operators (K, J) are introduced. Thus we proved the soliton hierarchy possesses the Bi-Hamilton structures and Liouville integrability. Moreover, with the help of Riccati equation, an infinite number of conservation laws for the soliton hierarchy are deduced; Finally, by using the method of derivation of functional under some constraint conditions, a complete one-to-one correspondence between the Hamiltonian functions of the hierarchy and its conservation density functions can be built.